text:
Axis of Evil (Unabridged)
Bobby Akart
The Axis of Evil had the perfect strategy.
Destroy the heartbeat of America.
Invade with their million man army.
But there was one thing they didn't count on...Texas.
Once again, political suspense collides with post-apocalyptic thriller fiction as sixteen-time bestselling author, Bobby Akart (The Blackout Series, The Pandemic Series, The Boston Brahmin Series and the Prepping for Tomorrow Series) intertwines historical fact with plausible fiction in The Lone Star Series.
For decades, American administrations have forced our military to be neutered in many respects - hampered by restrained rules of engagement, passed in strength by Russia and China, and tested by rogue nations like North Korea and Iran. As a result, nuclear Armageddon hangs over us like a mighty sword.
Military analysts agree that the United States can't start World War III. Why? We might lose. Sabers are rattling and the war of words escalate between America and her enemies, but the rhetoric has done little to stop the nuclear proliferation of North Korea and Iran. When the deep state, a close-knit group of powerful, influential people within our government challenges a new, untested President, things go awry.
The decisions made in the darkened rooms of Washington have a profound impact on ordinary Americans like the Armstrong family, Texas ranchers who spend their time making a living, but also preparing for the worst. As most of the family lives their life in Texas, one of them will unknowingly be the catalyst for World War III.
About the Lone Star Series
Political suspense collides with post-apocalyptic thriller fiction with the Lone Star Series by Bestselling Author Bobby Akart. A devastating EMP attack thrusts most of America into darkness, except for one state which has taken the steps to protect its power grid Texas.
World War III is imminent and a new President faces terrible choices as she navigates the shark-infested waters of Washington, and the geopolitical complexities around the world.
From Washington to Austin to a ranch in West Texas, decisions are made to protect the lives of Americans, some of which are considered selfish, rather than for people s survival.
The Lone Star series is designed to be the most thought-provoking post-apocalyptic series ever written. It will be a reminder to us all that...
One day, tomorrow won't come.
Books in the Lone Star Series Axis of Evil - February 20, 2018 Beyond Borders - March, 2018 Lines in the Sand - April, 2018 Texas Strong - May, 2018 Fifth Column - June 2018 Suicide Six - July 2018
Sean Runnette
Deviltx1966 , 04/19/2021
Very Good!!!
Well written and accurate. I am one county south of Borden County. Very impressed with the accuracy of this area. Strongly suggest this book!
Beyond Borders (Unabridged)
Lines in the Sand (Unabridged)
Texas Strong (Unabridged)
Fifth Column (Unabridged)
Suicide Six (Unabridged)
Yellowstone: Survival: A Post-Apocalyptic Survival Thriller: The Yellowstone Series, Book 4 (Unabridged)


This mouse is very ergonomical and easy to use. The vertical position of your hand makes this mouse very comfortable for those who use the computer intensively. Also it works very good for users with arm-, wrist- and shoulder problems.
This is a wireless Grip-mouse. This mouse includes a USB-receiver/connector for your PC. More than 20% discount!
Vertical mouse with 5 buttons. The ergonomic position of the hand reduces strain on your hand, fingers and arm.


The Hampshire Country & Garden Festival returns to Bere Mill for 2019. A fantastic day out for all the family! The event is held in support of the North Hampshire Medical Fund, buying much-needed equipment for Basingstoke, Winchester & Andover Hospitals.
The event will start with the Opening Evening on Saturday 8th June 7:30 – 10:00pm, and the main festival will be held on Sunday 9th June 2019 10am – 5pm.
For more information and to buy tickets visit the Hampshire & Country Garden Festival page.


text:
Belgium - Gross Capital Formation (% Of GDP)2022 Data 2023 Forecast 1970-2020 Historical
Gross capital formation (% of GDP) in Belgium was reported at 24.16 % in 2020, according to the World Bank collection of development indicators, compiled from officially recognized sources. Belgium - Gross capital formation (% of GDP) - actual values, historical data, forecasts and projections were sourced from the World Bank on January of 2022.

source: tradingeconomics.com
Gross capital formation (formerly gross domestic investment) consists of outlays on additions to the fixed assets of the economy plus net changes in the level of inventories. Fixed assets include land improvements (fences, ditches, drains, and so on); plant, machinery, and equipment purchases; and the construction of roads, railways, and the like, including schools, offices, hospitals, private residential dwellings, and commercial and industrial buildings. Inventories are stocks of goods held by firms to meet temporary or unexpected fluctuations in production or sales, and "work in progress." According to the 1993 SNA, net acquisitions of valuables are also considered capital formation.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: C# - How to add X minutes to current UNIX timestamp This is how I'm getting my current UNIX timestamp.
Int32 unixTimestamp = (Int32)(DateTime.UtcNow.Subtract(new DateTime(1970, 1, 1))).TotalSeconds;

How can I add on 10 minutes on this current UNIX timestamp, and have it still be formatted as a UNIX timestamp.
Thank you in advance.

A: Use AddMinutes(x).
Int32 unixTimestamp = (Int32)(DateTime.UtcNow.AddMinutes(10).Subtract(new DateTime(1970, 1, 1))).TotalSeconds;

In php:
<?php
    $inTenMinutes = time() + 10 * 60;
?>

redpajama_set_name: RedPajamaStackExchange


Being a part of the Global Entrepreneurship Week was not just a dream come true for me; but more of an eye opener. This is the place that I was sure I was going to get someone to invest in my business services. However, because it needed a lot of money, so that it could go to the heights that I envisioned starting out. The journey to get my business services where it right now has not been easy. Also, I have had to face a lot of challenges; that getting to this point has been. Now as I see it, the easy part.
Anyways, I came up with my business services tips and ideas while I was still in college. Also, I tried getting funding so that I could kick start it but to no avail. I tried a lot of banks. But there was none simply brave enough to invest in a nineteen-year-olds dream. Luckily for me, I had patented my idea. So I was sure that there was no way that anyone would steal it from me. This is a problem that a lot of brilliant young minds have faced.
Suddenly, before you know it, they see it on TV. Some big corporation is also claiming that it is their own. If they hadn't patented it, there is no way that you can be compensated. Let alone be heard in court. The ;fact that patenting requires some legal fees as well. In order for it to be done, which has also scared off a lot of young minds to abandoning their projects.
Therefore, I was called and told that I was going to be part of the brilliant young minds. That have been chosen to represent my country at this event. I was more than overjoyed. However, I was anxious at the same time. The reason was because I knew that this is exactly the kind of platform; that I had been waiting for to sell my idea to the whole world. I deal in business services; that helps disabled people access any medicine or healthcare that they need easily.
That was brought about by having grown up in a family that had a younger brother who was disabled. It hurt me so much seeing him going up like that. Most times he couldn't even access the right kind of medication that he needed. Because of this, we did not know where to get it. This is what gave me the idea for that. Furthermore, I developed this incredible application that they can use on their mobile phones; to get the medicine. Or perhaps, any doctor that they needed that is close to them.
I remember looking for a chiropractor in the next town; which was about twenty miles from where we lived. While there was one in our hometown; but we had no idea. It is this kind of sentiment that made me develop this app. So far, I am pleased with the kind of response that it has gotten. This is what prompted the call from the Organizers at GEW.
With that said, I have helped hundreds of people with this app. But I'm sure that if I get the right kind of funding; I will put it out there. Millions all across the world would get access to it. This is why this opportunity is just the best thing that has happened in my life. Although I must say; that getting accepted to the college of my dreams was also something that was quite big. Not till I got there and found out that there was nothing special about it. All we did was mostly have parties. And shout in the streets at night.
Under any circumstances, I am glad that I may atone for the few wrongs that I did on those nights. Thanks to this application that is helping hundreds of people. Before the end of the event; I had gotten a lot of proposals from different organizations; that were willing to invest in my idea. Some even showed me in the span of a few minutes how we would go about the marketing. As well as making the application even better. I am glad that I got to be a part of this. And I hope that much more brilliant minds get this opportunity too. As there are a lot of bright ideas out there; that could be very profitable.


Our unique weekend long event has grown over the years as a village-type and family-friendly festival.
The AFRICAN VILLAGE ARTS FESTIVAL is held each year the third Saturday in August and provides a space for artists and art lovers of every age and background to come together, create and generally enjoy the spirit of a fun, inclusive, well-organized community arts festival.
The Festival's name gives honor to those who established or still maintain families, homes and businesses in Olde Towne East, a historic area famed for being one of the city's first neighborhoods. The name is also meant to represent the sense of community, or village, the diverse artists and neighbors have extended to the William H. Thomas Art Gallery for more than 30 years. Each year one of Columbus' finest artists designs the Festival artwork.
…as well as Story Telling on the stage at African Village Arts Festival 2013 (this photo by KOYO, all other photos on this website by Sabine Lampe) We say THANK YOU to all our volunteers, board members, visitors, friends, and sponsors!


Perform as a teacher in undergraduate and graduate, in clinical psychology and related subjects. Carry out consultancy and extension work specific to the field of psychology, as well as interdisciplinary. Participate in academic and internal and external disciplinary meetings, as well as in the academic-administrative activities of the Department. Develop research in its area of knowledge, and promote the national and international positioning of the activities of the Department of Psychology.


text:
Taiwanese firefighters train in Pa., visit York City
Firefighters from Taiwan stopped by to trade stories about working in their respective countries after training at the HACC fire academy.
Taiwanese firefighters train in Pa., visit York City Firefighters from Taiwan stopped by to trade stories about working in their respective countries after training at the HACC fire academy. Check out this story on yorkdispatch.com: https://www.yorkdispatch.com/story/news/2017/12/08/taiwanese-firefighters-train-pa-visit-york-city/928445001/
Lindsay C. VanAsdalan, 505-5450/@lcvanasdalan Published 2:19 p.m. ET Dec. 8, 2017 | Updated 5:32 p.m. ET Dec. 11, 2017
Eight firefighters from Taiwan visit the York City fire department after training at the HACC fire academy in Harrisburg from Saturday, Dec. 2 to Wednesday, Dec. 6. (Photo c/o the York City fire department)(Photo: Submitted)
York City's fire department had some international visitors recently.
Firefighters from Taiwan stopped by to trade stories about working in their respective countries after training Dec. 2-6 in Harrisburg at the Harrisburg Area Community College (HACC) fire academy.
More: VIDEO: Firefighters save Lower Chanceford home
York City Deputy Fire Chief Chad Deardorff was lead instructor, providing rapid intervention training — which involves a team of firefighters dedicated solely to the firefighters inside a structure.
"That is something they do not have in Taiwan, " Deardorff said. "Two years ago, they had six firefighters killed in a collapse."
Deardorff said the Taiwanese firefighters were doing some research and saw studies of how the training had been successful.
They knew a fellow firefighter, Hsien-Yu Chu, from their country who had taken classes with HACC, and he helped connect them with the program.
Eight came for the training, and Chu accompanied them as a translator.
Eight firefighters from Taiwan visit the York City fire department after training at the HACC fire academy in Harrisburg from Saturday, Dec. 2 to Wednesday, Dec. 6. (Photo c/o the York City fire department) (Photo: Submitted)
"It was good to compare some stories and hear what they go through," said York City Fire Chief David Michaels.
He said he learned about some differences in the fire-service culture.
American firefighters have traditional-looking helmets and gear, while the Taiwanese have more of a modern look, he said, and their buildings are constructed differently, which affects how they work.
Deardorff explained that Taiwanese buildings are all concrete and metal, so the structure won't burn, even if what's inside will.
Because there's a lower risk of collapse, their firefighters can take time for a more methodical search of the building rather than doing a quick search first and going back a second time for a more-involved search, as American firefighters do.
But they also cannot cut holes in their metal roofs for vertical ventilation.
"We showed them some things, and we learned some things," Deardorff said.
The Americans learned about different patterns of going through buildings and how firefighters in Taiwan would deploy their equipment.
"It does open your eyes up to different things that are out there — not just in our country but internationally," Michaels said.
More: York City, firefighters strike deal on five-year contract
The visiting firefighters also participated in a structural burn session and live fire training, a "mayday situation" to get a firefighter out of building, Deardorff said.
"They finished out (the training) strong and did a great job," he said, adding that they had a lot of fun and were hard workers.
"It was kind of a sad moment knowing we won't see them tomorrow," he said. "(It was) maybe a once-in-a lifetime experience."
Read or Share this story: https://www.yorkdispatch.com/story/news/2017/12/08/taiwanese-firefighters-train-pa-visit-york-city/928445001/
Some York County municipalities served by Penn Waste weren't aware of sale
School tax caps in York County highest in a decade. See how high your district can go
The storm forecast has changed. See how much snow York County can expect
York City Police: Man shot in stomach Saturday
Local obituaries for Monday, Jan. 20
Police: Woman injured in stabbing early Saturday in York City


text:
\section{Introduction}

Density dependent Markov population processes, in which the transition rates depend
on the density of individuals in the population, have proved widely useful as models
in the social and life
sciences: see, for example, the monograph of Kurtz~(1981), in which approximations in terms of
diffusions are extensively discussed, in the limit as the typical population size~$n$ tends to
infinity. In the present paper we consider local approximation to their equilibrium
distributions~$\Pi_n$. In Socoll~\&~Barbour~(2008) [SB], total variation approximation
to~$\Pi_n$ by a suitably translated Poisson distribution was shown to be accurate to
order $O(n^{-\alpha/2})$, provided that the jump distributions satisfy a $(2+\alpha)$-th moment condition
for some $0 < \alpha \le 1$. Here, we examine the approximation of point probabilities by
those of the same translated Poisson distribution, and show
in Theorem~\ref{limth} that, under the same
assumptions, the error is now of order $O(n^{-(\alpha+1)/2}\sqrt{\log n})$. This is only
worse by the logarithmic factor than the best that can be obtained under comparable
conditions for sums of independent integer valued random variables.

A key ingredient in the proof of total variation approximation in~[SB] was to show
that the total variation distance between~$\Pi_n$ and its unit translate $\Pi_n*\delta_1$
is of order~$O(n^{-1/2})$.  Here, we need to establish a local limit analogue of
this theorem.  We prove in Section~2 that the differences between the point
probabilities of~$\Pi_n$ and those of its unit translate are uniformly bounded by a
quantity of order $O(n^{-1}\sqrt{\log n})$. An important step in proving this is to
establish that, for some $U\ge1$, the difference between ${\mathbb P}[Z_n(t) = k+1 \,|\, Z_n(0)=i]$
and ${\mathbb P}[Z_n(t) = k \,|\, Z_n(0)=i-1]$ is of order  $O(n^{-1}\sqrt{\log n})$, uniformly
for~$i$ in a set~$I$ such that $\Pi_n(I^c) = O(n^{-1})$.  This is achieved by a
pathwise comparison of probability densities, and using a martingale concentration
inequality.  Note that, for sums of independent random variables, the corresponding
difference is always zero, so that this problem does not arise there.

The proof of Theorem~\ref{limth} is undertaken in Section~3.  The argument relies on
the Stein--Chen method (Chen, 1975) and Dynkin's formula, exploiting the particularly nice
properties of the solutions to the Stein--Chen equation for one point subsets of~$\mathbb Z_+$.

\subsection*{Preliminaries}\label{prelims}

For each $n\in\mathbb N$, let $Z_{n}(t)$, $t\ge0$,  be an
irreducible continuous time pure jump Markov process taking values in $\mathbb Z$, with transition
rates given by
$$
  i\ \to\  i+j \quad \mbox{ at rate }\quad n\lambda_j\Big(\frac{i}{n}\Big),\qquad i \in {\mathbb Z},\
  j\in \mathbb Z\setminus \{0\},
$$
where the $\lambda_j(\cdot)$ are prescribed functions on $\mathbb R$.
We then define the `overall jump rate' of the process $n^{-1}Z_{n}$ at
$z \in n^{-1}{\mathbb Z}$ by
$$
  \Lambda(z) \ :=\ \sum_{j\in \mathbb Z\setminus \{0\}} {{\lambda}_{j}}(z),
$$
its `average growth rate' by
$$
  F(z)\ :=\ \sum_{j\in \mathbb Z\setminus \{0\}}j{{\lambda}_{j}}(z),
$$
and its `quadratic variation' function by $n^{-1}{\sigma}^2(z)$, where
$$
  {\sigma}^2(z)\ =\ \sum_{j\in \mathbb Z\setminus \{0\}} j^2  {{\lambda}_{j}}(z),
$$
assumed to be finite for all~$z\in{\mathbb R}$.

We make the following assumptions on the functions ${\lambda}_{j}$; they
are discussed at greater length in~[SB].
\\[1ex]
{\bf \small A1:} There exists a unique $c$ satisfying $F(c)=0$; furthermore, $F'(c) < 0$
    and, for any $\eta>0$, $\mu_{\eta}:= \inf_{|z-c|\geq {\eta}}|F(z)|>0$.
\\[1ex]
{\bf \small A2:}
   For each $j\in {\mathbb Z}\setminus \{0,\}$, there exists $c_j\ge0$ such that
    \begin{equation} \label{lamdaj}
      \lambda_j(z)\leq c_j(1+|z-c|), \qquad z\in \mathbb R,
    \end{equation}
    where the~$c_j$ are such that, for some $0 < \alpha \le 1$,
    $$
       \sum_{j \in \mathbb Z \setminus \{0\}}|j|^{2+\alpha}c_{j} =: s_\alpha <\infty.
    $$
\ignore{
The moment condition on the~$c_j$ in Assumption~A2\,(a) plays the same r\^ole as
the analogous moment condition in the Lyapounov central limit theorem.  Under
this assumption, the ideal rate of convergence in the usual central limit
approximation is the rate~$O(n^{-\alpha/2})$ that we had established for our total
variation approximation.
Assumption A2\,(b) is important for establishing the smoothness of the equilibrium
distribution~$\Pi_n$.  If, for instance, all jump sizes were multiples of~$2$, the
approximation that we are concerned with would not be accurate in total variation.
}
{\bf \small A3:}
  There exist $\varepsilon>0$ and $0 < \delta \le 1$ and a set $J \subset
     {\mathbb Z}\setminus \{0\}$ with $1 \in J$ such that
     \begin{eqnarray*}
       &&\inf_{|z-c|\leq \delta}\lambda_j(z)\ \geq\ \varepsilon \lambda_j(c) \ > 0,\ \ \ j\in J;\\
       &&\lambda_j(z)=0 \  \mbox{for all}\   |z-c|\leq \delta,\ \ \ j\notin J.
     \end{eqnarray*}
\ignore{
Assumptions A2\,(a) and A3 imply in particular that the series
$\sum_{j \in \mathbb Z \setminus \{0\}}j\lambda_j(z)$ and
$\sum_{j \in \mathbb Z \setminus \{0\}}j^2\lambda_j(z)$  are uniformly convergent
on~$|z-c| \le \delta$,  and that their sums,
$F$ and ${\sigma}^2$ respectively, are continuous there. They also imply that
$$
  \sum_{j\in \mathbb Z\setminus \{0\}}|j|n \lambda_j(i/n)
     \ =\  O(|i|),\ \ \ |i|\to \infty,
$$
so that the process $Z_n$ is a.s.\ non-explosive, in view of
Hamza \& Klebaner~(1995, Corollary~2.1).
}
\\[0.1ex]
{\bf \small A4:}
  \ (a)\ For each $ j\in J$, ${\lambda}_{j}$  is of class $C^2$ on~$|z-c| \le \delta$.\\[0.3ex]
\mbox{} \qquad  (b)\  For $\delta$ as in A3,
$$
   L_1 \ :=\
    \sup_{j\in J}\frac{\|{{\lambda}_j^{\prime}}\|_{\delta}}{\lambda_j(c)}
      \ <\ \infty;\qquad
    L_2\ :=\
   \sup_{j\in J} \frac{\|{{\lambda }_j^{\prime \prime}}\|_{\delta}}{|j|\lambda_j(c)}
        \ <\ \infty,
$$
where   $\|f\|_{\delta}:=\sup_{|z-c|\leq \delta}|f(z)|$. \\[0.4ex]
\ignore{
This assumption implies in particular, in view of Assumptions A2--A3, that the series
$\sum_{j \in \mathbb Z \setminus \{0\}}j\lambda^{\prime}_j(z)$ and
$\sum_{j \in \mathbb Z \setminus \{0\}}j^2\lambda^{\prime}_j(z)$ are uniformly
convergent on $|z-c|\leq \delta$,
that their sums are
$F^{\prime}$ and~$(\sigma^2)^{\prime}$  respectively, and that
$F$ and $\sigma^2$  are of class $C^1$ on $|z-c|\leq \delta$.
}
\ignore{
This assumption implies, in view of A2--A3, that the series
$\sum_{j \in \mathbb Z \setminus \{0\}}j\lambda^{\prime \prime}_j(z)$
is uniformly convergent on $|z-c|\leq \delta$, its sum is
$F^{\prime \prime}$,
and~$F$ is of class $C^2$ on $|z-c|\leq \delta.$}

In [SB], it is shown that, under assumptions A1--A4,
\begin{equation}\label{dtv-thm}
  d_{TV}(\widehat{\Pi}_n, \widehat{\rm Po}(nv_c)) \ =\  O(n^{-\alpha/2}),
\end{equation}
where $v_c := \sigma^2(c)/\{-2F'(c)\}$, $\widehat{\rm Po}(nv_c)$ denotes the centred Poisson distribution
$$
  \widehat{\rm Po}(nv_c)\ :=\ {\rm Po}(nv_c)*\delta_{-\lfloor nv_c\rfloor},
$$
and $\widehat{\Pi}_n$ denotes the centred
equilibrium distribution ${\Pi_n}*\delta_{-\lfloor nc\rfloor}$;
here, $\delta_r$ denotes the point mass at~$r$, and~$*$ denotes convolution.

In this paper we prove the following complementary local limit approximation.

\begin{theorem} \label{limth} Under Assumptions A1--A4, there exists a
constant $C>0$ such that
 $$
  \sup_{k\in \mathbb Z} \Bigl|\widehat{\Pi}_{n}(k)-\widehat{\rm Po}(nv_c)\{k\}\Bigr|\ \le\
     C n^{-(\alpha+1)/2}\sqrt{\log n}.
$$
\end{theorem}
\noindent This theorem shows that, even at the level of point probabilities, the approximation
to~$\Pi_n*\delta_{-\lfloor nc\rfloor}$ provided by the centred Poisson distribution
$\widehat{\rm Po}(nv_c)$ is almost exactly the best that could be expected.

The proof is based on exploiting the equation
\begin{equation}\label{DF}
  {\mathbb E}({\mathcal A}_n h)(Z_{n})=0,
\end{equation}
where ${\mathcal A}_n$ denotes the infinitesimal generator of~$Z_{n}$:
$$
  ({\mathcal A}_n h)(i)\ :=\
   \sum_{j\in \mathbb Z\setminus \{0\}}n{{\lambda}_{j}}\Big(\frac{i}{n}\Big)\big[h(i+j)-h(i)\big],
\quad i \in {\mathbb Z},
$$
and where, here and subsequently, the quantity $Z_n$, when appearing without a time argument,
is to be interpreted in such expressions as being a random variable having the
equilibrium distribution~$\Pi_n$.  The equation~\Ref{DF} is a manifestation of
Dynkin's formula, and it holds under rather mild conditions on~$h$: see
Hamza \& Klebaner~(1995).

Manipulations carried out in [SB] show that ${\mathcal A}_n h$ can be expressed in alternative form.

\begin{lemma}{\rm [SB, Lemma 1.1.]} \label{generator}
Suppose that $\sigma^2(z) < \infty$ for all~$z\in{\mathbb R}$.
Then, for any function $h\colon \mathbb Z\to \mathbb R$  with bounded differences,  we have
\begin{equation}\label{basic}
   ({\mathcal A}_nh)(i)
      \ =\ \frac{n}{2}{\sigma}^2\Big(\frac{i}{n}\Big)\bigtriangledown{g_h}(i)
        +nF\Big(\frac{i}{n}\Big)g_h(i) + E_n(g_h,i), \nonumber
\end{equation}
where $\bigtriangledown{f}(i):=f(i)-f(i-1)$ and $g_h(i) := \bigtriangledown{h}(i+1)$
and, for any $i\in{\mathbb Z}$,
\begin{eqnarray}
   \lefteqn{E_n(g,i)}\nonumber\\
  &:=&  -\frac{n}{2}F\Big(\frac{i}{n}\Big)\bigtriangledown{g}(i)
      +\sum_{j\geq 2}a_j(g,i) n\lambda_j\Big(\frac{i}{n}\Big)
      -\sum_{j\geq 2}b_j(g,i) n{{\lambda}_{-j}}\Big(\frac{i}{n}\Big),
    \label{En-def} \nonumber
\end{eqnarray}
with
\begin{eqnarray}
  a_j(g,i) &:=& -\binom{j}{2}\bigtriangledown {g}(i)
     + \sum_{k=1}^{j-1} {k \bigtriangledown {g}(i+j-k)} \label{aj-bnd-1} \nonumber \\
  &=&   \sum_{k=2}^j \binom{k}{2} \bigtriangledown^2{g}(i+j-k+1);\label{aj-bnd-2} \nonumber \\
  b_j(g,i) &:=& \binom{j}{2}\bigtriangledown {g}(i)
     - { \sum_{k=1}^{j-1} k \bigtriangledown {g}(i-j+k)} \nonumber \\
  &=&   \sum_{k=2}^j \binom{k}{2} \bigtriangledown^2{g}(i-j+k). \nonumber
\end{eqnarray}
\end{lemma}

Since $F(c)=0$, we note that, for $i/n$ small, $\{-F'(c)\}^{-1}({\mathcal A}_n h)(i + {\lfloor nc \rfloor})$
is close to
\[
    \frac1{-F'(c)}\,\frac n2\,\sigma^2(c) \Delta g^*_h(i) - (i-{\langle nv_c \rangle})g^*_h(i)
		\ =\  nv_c \Delta g^*_h(i) - (i-{\langle nv_c \rangle})g^*_h(i),
\]
for $g^*_h(i) := g_h(i+{\lfloor nc \rfloor})$, where ${\langle nv_c \rangle} = nv_c - {\lfloor nv_c \rfloor}$ denotes the fractional
part of~$nv_c$.  This is the Stein operator for the centred Poisson distribution
$\widehat{\rm Po}(nv_c)$ (R\"ollin,~2005), acting on the function~$g^*_h$. Combining this observation
with~\Ref{DF} and writing $Y_n=Z_n-\lfloor nc \rfloor$ yields
\begin{eqnarray}
  0 &=& \{-F'(c)\}^{-1}{\mathbb E}\{({\mathcal A}_n h)(Y_n + {\lfloor nc \rfloor})\} \nonumber\\
    &=& {\mathbb E}\{nv_c\Delta g^*_h(Y_n) - (i-{\langle nv_c \rangle})g^*_h(Y_n)\} + {\mathbb E}\{H(g^*_h,Y_n)\},
		\label{nearly-Stein}
\end{eqnarray}
say. If the error term ${\mathbb E}\{H(g^*_h,Y_n)\}$ can be controlled,
then Stein's method leads easily to the approximation of
${\mathcal L}(Y_n) = \Pi_n*\delta_{-{\lfloor nc \rfloor}}$ by $\widehat{\rm Po}(nv_c)$.
For the approximation of point probabilities, \Ref{nearly-Stein} needs to be
analyzed for
functions~$g^*_h$ that are translates of the solutions to the Stein--Chen
equation corresponding to single point sets.

Carrying out this recipe, and examining the form of $H(g^*_h,Y_n)$, yields
\begin{eqnarray}\nonumber
  \lefteqn{{\sup_{r\in\integ}}|(\Pi_{n}-\lfloor nc \rfloor)(r)-\widehat{{\rm Po}}(n v_c)(r)|} \\\nonumber
  &&\ \le\  \frac{1}{-F^{\prime}(c)} {\sup_{r\in\integ}}\; |{\mathbb E} R(n,r;Y_n)|
      +{\sup_{r\in\integ}}\;n v_c\; |{\mathbb E}\{ \bigtriangledown^2{\tg_{nv_c,r}(Y_{n}+1)}\}|\\\nonumber
  &&\qquad\mbox{}+{\sup_{r\in\integ}}\;\widehat{{\rm Po}}(n v_c)\{r\}\cdot
     {\mathbb P}(Y_n< -\lfloor n v_c \rfloor)\\
  &&\ :=\ R_{n1}+R_{n2}+R_{n3}, \label{ek}
\end{eqnarray}\\
say, where
\begin{eqnarray}
  R(n,r;Y_n)&:=& \frac{n}{2}\Bigl[{\sigma}^2\Bigl(\frac{Y_n+\lfloor nc \rfloor}{n}\Bigr)
     -\sigma^2(c)\Bigr]\bigtriangledown{\tg_{nv_c,r}(Y_n)} \nonumber \\
   &&\quad\mbox{} +n\Bigl[F\Bigl(\frac{Y_n+\lfloor nc \rfloor}{n}\Bigr)-F(c)
       -\frac{Y_n}{n}F^{\prime}(c)\Bigr]\tg_{nv_c,r}(Y_n) \nonumber \\
   &&\quad\mbox{} + F^{\prime}(c)\langle nv_c \rangle \tg_{nv_c,r}(Y_n) + E_n(\tg_{nv_c,r},Y_n+{\lfloor nc \rfloor}),
   \label{rt}
\end{eqnarray}
and the function~$\tg_{nv_c,r}$ is given by
\begin{equation}\label{defgt}
  \tg_{nv_c,r}(l)\ :=\ \left\{\begin{array}{ll}
       0,     &{\rm if}\;\;\; l < -\lfloor n v_c \rfloor \\
       g_{n v_c, \{r+\lfloor n v_c\rfloor\}}(l+\lfloor n v_c \rfloor),
          &{\rm if} \;\;\; l\geq -\lfloor n v_c \rfloor.
       \end{array} \right.
\end{equation}
Here, for $A \subset {\mathbb Z}_+$, $g_{\mu,A}$ denotes the solution to the
Stein--Chen equation
\begin{equation}\label{SE}
   1 \hspace*{-1mm} {\rm l}_A(i)-{\rm Po}(\mu)\{A\} \ =\  \mu\;g_{\mu,A}(i+1)-i\;g_{\mu,A}(i), \qquad i\geq 0.
\end{equation}
We further split the last term of~\Ref{rt} into
\[
   E_n(\tg_{nv_c,r},Y_n+{\lfloor nc \rfloor}) \ =\  \sum_{l=1}^7 E_{nl}(\tg_{nv_c,r},Y_n+{\lfloor nc \rfloor}),
\]
with
{\allowdisplaybreaks
\begin{eqnarray}
  \label{En1}
    E_{n1}(\tg_{nv_c,r},Y_n+{\lfloor nc \rfloor}) &:=& -\frac{n}{2}\Bigl[F\Bigl(\frac{Y_n+\lfloor nc \rfloor}{n}\Bigr)
     -F(c)\Bigr]\bigtriangledown\tg_{nv_c,r}(Y_n) ; \\
  E_{n2}(\tg_{nv_c,r},Y_n+{\lfloor nc \rfloor})  &:=&   \sum_{j=2}^{{\lfloor \sqrt n \rfloor}} \Bigl[\sum_{k=2}^{j}\binom{k}{2}\bigtriangledown^2
       \tg_{nv_c,r}(Y_n+j-k+1)\Bigr]\,n \lambda_j(c); \label{En2} \\
   E_{n3}(\tg_{nv_c,r},Y_n+{\lfloor nc \rfloor})  &:=&  \sum_{j=2}^{{\lfloor \sqrt n \rfloor}} \Bigl[-\binom{j}{2}\bigtriangledown\tg_{nv_c,r}(Y_n)
        +\sum_{k=1}^{j-1} k \bigtriangledown\tg_{nv_c,r}(Y_n+j-k) \Bigr]\nonumber\\
     &&\mbox{}\qquad n \Bigl\{\lambda_j\Bigl(\frac{Y_n+\lfloor nc \rfloor}{n}\Bigr) - \lambda_j(c)\Bigr\};
        \label{En3} \\
   E_{n4}(\tg_{nv_c,r},Y_n+{\lfloor nc \rfloor})  &:=&  \sum_{j\ge {\lceil \sqrt n \rceil}}\Bigl[-\binom{j}{2}\bigtriangledown\tg_{nv_c,r}(Y_n)+\sum_{k=1}^{j-1} k
       \bigtriangledown\tg_{nv_c,r}(Y_n+j-k)\Bigr]\nonumber\\
      &&\mbox{}\qquad    n \lambda_j\Bigl(\frac{Y_n+\lfloor nc \rfloor}{n}\Bigr); \label{En4} \\
   E_{n5}(\tg_{nv_c,r},Y_n+{\lfloor nc \rfloor})  &:=&  -\sum_{j=2}^{{\lfloor \sqrt n \rfloor}}
     \Bigl[\sum_{k=2}^{j}\binom{k}{2}\bigtriangledown^2\tg_{nv_c,r}(Y_n-j+k)\Bigr]\,n\lambda_{-j}(c); \label{En5}\\
   E_{n6}(\tg_{nv_c,r},Y_n+{\lfloor nc \rfloor})  &:=&  -\sum_{j=2}^{{\lfloor \sqrt n \rfloor}} \Bigl[\binom{j}{2}\bigtriangledown\tg_{nv_c,r}(Y_n)
     -\sum_{k=1}^{j-1} k \bigtriangledown\tg_{nv_c,r}(Y_n-j+k)\Bigr]\nonumber\\
     &&\mbox{}\qquad n
       \Bigl\{\lambda_{-j}\Bigl(\frac{Y_n+\lfloor nc \rfloor}{n}\Bigr) - \lambda_{-j}(c)\Bigr\};
			 \label{En6} \\
   E_{n7}(\tg_{nv_c,r},Y_n+{\lfloor nc \rfloor})  &:=&  - \sum_{j\ge {\lceil \sqrt n \rceil}}
	   \Bigl[\binom{j}{2}\bigtriangledown\tg_{nv_c,r}(Y_n)-\sum_{k=1}^{j-1} k \bigtriangledown\tg_{nv_c,r}(Y_n-j+k)\Bigr]\nonumber\\
     &&\mbox{}\qquad n\lambda_{-j}\Bigl(\frac{Y_n+\lfloor nc \rfloor}{n}\Bigr).
         \label{En7}
\end{eqnarray}
}
Our strategy for proving Theorem~\ref{limth} is now to show that each of the terms
$R_{n1}$, $R_{n2}$ and $R_{n3}$ in~\Ref{ek} is
of the desired order $O(n^{-(\alpha+1)/2}\sqrt{\log n})$; clearly, the treatment of~$R_{n1}$, which
involves all the detail of~$E_n(\tg_{nv_c,r},Y_n+{\lfloor nc \rfloor})$, is to be the most laborious.

 \setcounter{equation}{0}
 \setcounter{theorem}{0}
\section[Point probabilities]{Differences of point probabilities}
As an essential step in proving Theorem~\ref{limth}, we need first to show that the differences
between the successive point probabilities of~$\Pi_n$ are suitably small.
The bound that we achieve is of
order $O(n^{-1}\sqrt{\log n})$.  In order to prove this result, we begin with
two lemmas.  The first states that, for any $U\ge1$, the distribution of~$Z_n(U)$
has point probabilities
which are uniformly of order $O(n^{-1/2})$, if~$Z_n(0)$ is close enough to~$nc$.

\begin{lemma}\label{sdoi}
Under Assumptions A1--A4, for any $U\ge1$, there exists $C_{\ref{sdoi}}(U) < \infty$ such that
$$
  \sup_{k\in \mathbb Z} \sup_{|i-nc|\le n\delta/2}\; {\mathbb P}(Z_{n}(U)=k \mid
     Z_{n}(0)=i)\ \leq\ C_{\ref{sdoi}}(U)n^{-1/2}.
$$
\end{lemma}

\begin{proof}
 Note that, for any integer valued random variable~$X$,
\begin{eqnarray}
   \sup_{k\in{\mathbb Z}}\mathbb P(X=k)
	 &=& \sup_{k\in{\mathbb Z}}\{\mathbb P(X\leq k)-\mathbb P(X+1\leq k)\}
      \nonumber \\
   &\le& d_{TV}\{\mathcal{L}(X),\mathcal{L}(X)*\delta_1\}, \label{point-bnd}
\end{eqnarray}
where $\mathcal{L}(X)$ denotes the distribution of $X$. Taking $X = Z_n(U)$ and applying
Lemma~\ref{lema4} completes the proof.
\end{proof}

The next lemma shows that the {\em differences\/} between successive point probabilities
of~$Z_n(U)$ are uniformly close, to order~$O(n^{-1}\sqrt{\log n})$,
for a large range of values of~$Z_n(0)$ and for a particular choice of~$U\ge1$.
This is the result
that we shall then be able to extend to the equilibrium distribution~$\Pi_n$.
For $\Lambda^* := \sup_{|z-c| \le \delta/2} \Lambda(z)$, we set
\begin{equation}\label{U-def}
  U\ :=\ \max\{1,1/2\Lambda^*\};\qquad \delta'_1 \ :=\ \delta e^{-U\|F'\|_\delta}/4.
\end{equation}

\begin{lemma}\label{sunu}
Under Assumptions A1--A4, and for $U$ and~$\delta'_1$ defined above,
there exists $C_{\ref{sunu}}< \infty$ such that
\begin{eqnarray*}
  &&\sup_{k\in \mathbb Z}\sup_{|i-nc|\le n\delta'_1}
   |{\mathbb P}(Z_{n}(U)=k \mid Z_{n}(0)=i-1)-{\mathbb P}(Z_{n}(U)=k+1 \mid Z_{n}(0)=i)|\\
  &&\qquad \qquad \qquad \leq\ C_{\ref{sunu}}\ n^{-1}\sqrt{\log n}.
\end{eqnarray*}
\end{lemma}

\begin{proof}
We compare the probability measures ${\mathcal L}((Z_n(u),\,0\le u\le U) \,|\, Z_n(0) = i-1)*\delta_1$
and ${\mathcal L}((Z_n(u),\,0\le u\le U) \,|\, Z_n(0) = i)$ by examining the likelihood ratio of
the processes $Z_n^{(1)} =_d \{Z_n \,|\, Z_n(0)=i-1\}$ and $Z_n^{(2)} =_d \{Z_n \,|\, Z_n(0)=i\}$
 along paths with the same set of jumps $(j_l,\,l\ge1)$
occurring at the same times $(t_l,\,l\ge1)$.
$Z_n^{(1)}$ starts from the state~$i-1$; we write
$z_l := n^{-1}\{i-1+\sum_{s=1}^l j_s\}$ for the value of $n^{-1}Z_n^{(1)}$ at
time~$t_l$, $l\ge0$. $Z_n^{(2)}$ starts from the state~$i$, and thus has the same
paths as $Z_n^{(1)} + 1$.  Then
the likelihood ratio of the two processes along the first~$m$ steps of the path is given
by
\begin{eqnarray*}
  S_m &:=& S_m(z_0,z_1\ldots,z_m; t_1,\ldots,t_m)\\
  &=& \prod_{l=0}^{m-1} \left\{ \frac{\lambda_{j_{l+1}}(z_l + n^{-1})}{\lambda_{j_{l+1}}(z_l)}
   \exp\{-n(\Lambda(z_l+n^{-1}) - \Lambda(z_l))(t_{l+1} - t_l)\} \right\} \\
  &=&  \prod_{l=0}^{m-1} V_l.
\end{eqnarray*}
Note that, since $|(1+x)(1+y) - 1| \le 3|x| + |y|$
if $|y| \le 2$, and since $|e^t-1| \le 2|t|$ in $t \le 1$, it follows that
\[
  |V_l-1| \ \le\   3\Bigl|\frac{\lambda_{j_{l+1}}(z_l + n^{-1})}{\lambda_{j_{l+1}}(z_l)} - 1 \Bigr|
   + 2n|\Lambda(z_l+n^{-1}) - \Lambda(z_l)|(t_{l+1} - t_l)
\]
provided that
\[
    n\{\Lambda(z_l) - \Lambda(z_l+n^{-1})\}(t_{l+1} - t_l) \ \le\  1.
\]

Now, if $|z-c| \le \delta/2$ and $n^{-1} \le \delta/2$, it follows from Assumptions~A2
and~A4 that
\[
  \Bigl|\frac{\lambda_{j}(z + n^{-1})}{\lambda_{j}(z)} - 1 \Bigr|
  \ \le\  \frac{\| \lambda'_j\|_\delta}{n\varepsilon\lambda_j(c)} \ \le\  \frac{L_1}{n\varepsilon},
\]
and similarly that
\begin{equation}\label{L-ratio}
   \Bigl|\frac{\Lambda(z + n^{-1})}{\Lambda(z)} - 1 \Bigr|
  \ \le\  \frac{\| \Lambda'\|_\delta}{n\varepsilon\Lambda(c)} \ \le\  \frac{L_1}{n\varepsilon}
\end{equation}
also.  Hence, for all $n\ge2/\delta$, writing $e_{l+1} := n\Lambda(z_l)(t_{l+1} - t_l)$, we have
\begin{equation}\label{Ul-bnd}
  |V_l-1| \ \le\  \frac{L_1}{n\varepsilon}\{3 + 2e_{l+1}\},
\end{equation}
as long as
\begin{equation}\label{1st-conds}
  |z_l-c| \le \delta/2\quad \mbox{and either}\quad  \Lambda(z_l) \le \Lambda(z_l+n^{-1})
	     \quad \mbox{or}\quad e_{l+1} \le n\varepsilon/L_1.
\end{equation}

Now consider the random likelihood ratio process
\[
   (S_m(n^{-1}Z_0,n^{-1}Z_1,\ldots,n^{-1}Z_m;\tau_1,\ldots,\tau_m),\, m\ge0),
\]
where $(\tau_l,\,l\ge0)$ denote the jump times of the process~$Z_n^{(1)}$,
and $Z_l := Z_n^{(1)}(\tau_l)$, $l\ge0$, the sequence of states
that it visits; define also $E_{l} \ :=\ n\Lambda(n^{-1}Z_{l-1})(\tau_{l} - \tau_{l-1})$.
Then~$S := (S_m,\, m\ge0)$, is a martingale with mean~$1$ with respect
to the filtration ${\mathcal G}_m := \sigma(Z_0,Z_1,\ldots,Z_m;\tau_1,\ldots,\tau_m)$, $m\ge0$.  We shall,
for technical reasons, work rather with another martingale~${\widetilde S}$, which
typically agrees with~$S$
for a long time, but which satisfies the inequality
\begin{equation}\label{key-bnd}
  |{\widetilde S}_{m+1} - {\widetilde S}_m| \ \le\  \frac{2L_1}{n\varepsilon}\,\{3 + 2E_{m+1}\}
\end{equation}
for all~$m\ge0$.  This we achieve by defining
$\sigma := \min\{\sigma_r,\,1\le r\le 3\}$, where
\begin{eqnarray}
  \sigma_1 &:=& \inf\{l\ge0 \colon\, n|\Lambda(n^{-1}[Z_{l-1}+1]) - \Lambda(n^{-1}Z_{l-1})|(\tau_l - \tau_{l-1}) > 1\},
                   \label{sig1-def}\\
  \sigma_2 &:=& \inf\{l\ge0 \colon\, S_l > 2\} \quad \mbox{and} \quad
  \sigma_3 \ :=\ \inf\{l\ge0 \colon\, |n^{-1} Z_l - c| > \delta/2\},\label{sig23-def}
\end{eqnarray}
and then setting
\[
   {\widetilde S}_m \ :=\ S_{m\wedge\sigma} C_{m,\sigma_1},
\]
where
\[
    C_{m,l} \ :=\ \begin{cases}
       e/V_{l-1} &\mbox{if}\quad l \le \min\{m,\sigma_2,\sigma_3\}\ \mbox{and}\
			    \Lambda(z_{l-1}) > \Lambda(z_{l-1}+n^{-1});\\
		1 &\mbox{else}.
       \end{cases}
\]
Note that the only effect of the factor~$C_{m,\sigma_1}$ is to multiply
${\widetilde S}$ by~$e$ instead of by~$V_{\sigma_1-1}$ at time~$\sigma_1$, if $\sigma_1 \le \min\{\sigma_2,\sigma_3\}$
and $\Lambda(z_{\sigma_1-1}) > \Lambda(z_{\sigma_1-1}+n^{-1})$. The value~$e$ is chosen so that the
martingale property is preserved; and the modification also ensures that~\Ref{key-bnd}
is still satisfied at time~$\sigma_1$, since $2(e-1)$ is no larger that $4L_1 E_{\sigma_1}/\{n\varepsilon\}$,
because, at time $\sigma_1$,
\begin{eqnarray*}
    1  &<&  n|\Lambda(z_{\sigma_1-1}+n^{-1}) - \Lambda(z_{\sigma_1-1})|(\tau_{\sigma_1} - \tau_{\sigma_1-1}) \\
     &=& E_{\sigma_1}\Bigl|\frac{\Lambda(z_{\sigma_1-1}+n^{-1})}{\Lambda(z_{\sigma_1-1})} - 1\Bigr|
     \ \le\    E_{\sigma_1}\,\frac{L_1}{n\varepsilon},
\end{eqnarray*}
in view of~\Ref{L-ratio}.

Now, from~\Ref{key-bnd}, and since also, by the strong Markov property, the
conditional distribution ${\mathcal L}(E_{l+1} \,|\, {\mathcal G}_l)$ is the standard exponential
$\exp(1)$ distribution for each~$l$, the process~${\widetilde S}$ satisfies
the conditions of the variant of the bounded differences inequality for martingales
given in Barbour~(2008, Lemma~4.1), from which it follows that
\[
  {\mathbb P}\left[|{\widetilde S}_m - 1| > C\,\frac{L_1\sqrt{m\log m}}{n\varepsilon} \,\Big|\, Z_n(0) = i-1\right]
     \ \le\  2\exp\{-3C\log m/928\}
\]
for any $m$ such that
\[
     \sqrt{\frac m{\log m}} \ \ge\ 135 C / 236.
\]
In particular, recalling~\Ref{U-def}, for $m = m(n) := \lceil 2n\Lambda^*U \rceil$,
we have
\begin{equation}\label{R-tilde-bnd}
    {\mathbb P}\left[|{\widetilde S}_{m(n)} - 1| > C\,\frac{L_1\sqrt{m(n)\log m(n)}}{n\varepsilon} \,\Big|\, Z_n(0) = i-1\right]
            \ \le\  2n^{-3},
\end{equation}
if we take $C := 928$, as long as $n\ge e$ and
\begin{equation}\label{n-cond}
    \frac n{\log n} \ \ge\ 540^2.
\end{equation}

Now let $M_n(U) := \min\{l\colon\,\tau_l > U\}$, and introduce the notation ${\mathbb P}_s$
to denote ${\mathbb P}[\cdot\,|\, Z_n(0)=s]$. Then
\begin{eqnarray}
   \lefteqn{{\mathbb P}_{i-1}[\{M_n(U) > m(n)\} \cap \{\sigma_3 \ge M_n(U)\}] }\nonumber\\
     &&\ =\  {\mathbb P}_{i-1}[\{\tau_{m(n)} \le U\} \cap \{\sigma_3 \ge M_n(U)\}]\nonumber\\
     &&\ \le\  {\rm Po\,}(n\Lambda^*U)\{(2n\Lambda^*U,\infty)\} \ \le\  \exp\{-n\Lambda^*U/3\}, \label{chernoff}
\end{eqnarray}
by the Chernoff inequality (see Chung \& Lu (2006, Theorem~4)).  Hence, for~$U$ as defined
in~\Ref{U-def}, we have
\begin{eqnarray}
  \lefteqn{|{\mathbb P}_i[Z_n(U) = k+1 ] - {\mathbb P}_{i-1}[Z_n(U) = k]|
      \ =\  |{\mathbb E}_{i-1}\{(S_{M_n(U)} - 1)I[Z_n(U) = k]\}| }\nonumber\\
  &\le& |{\mathbb E}_{i-1}\{({\widetilde S}_{M_n(U)} - 1)I[Z_n(U) = k] I[{\widetilde S}_{M_n(U)} = S_{M_n(U)}]\}|
     \   +\ {\mathbb P}_{i-1}[{\widetilde S}_{M_n(U)} \neq S_{M_n(U)}]\nonumber\\
  &\le& |{\mathbb E}_{i-1}\{({\widetilde S}_{m(n)} - 1)I[M_n(U) \le m(N)]I[Z_n(U) = k] I[{\widetilde S}_{M_n(U)} = S_{M_n(U)}]
                      \}|\nonumber\\
    &&\mbox{}  + {\mathbb P}_{i-1}[\{{\widetilde S}_{M_n(U)} \neq S_{M_n(U)}\} \cap \{M_n(U) \le m(N)\}]
                + {\mathbb P}_{i-1}[M_n(U) > m(n)].\label{half-done}
\end{eqnarray}
Applying \Ref{R-tilde-bnd} and~\Ref{chernoff}, \Ref{half-done} now implies that, for
all~$n \ge e$ satisfying~\Ref{n-cond},
\begin{eqnarray}
   \lefteqn{|{\mathbb P}_i[Z_n(U) = k+1] - {\mathbb P}_{i-1}[Z_n(U) = k]|}\nonumber\\
   &\le& \frac{CL_1}{n\varepsilon} \sqrt{m(n)\log m(n)}\, {\mathbb P}_{i-1}[Z_n(U)=k] + 2n^{-3}
        + {\mathbb P}_{i-1}[\cup_{l=0}^4 A_l] ,
       \label{nearly}
\end{eqnarray}
where
\begin{eqnarray*}
   A_0 &:=& \left\{|{\widetilde S}_{m(n)} - 1| > C\,\frac{L_1\sqrt{m(n)\log m(n)}}{n\varepsilon}\right\};
      \qquad   A_1 := \{\sigma_1 \le M_n(U)\};\\
   A_2 &:=& \{\sigma_2 \le M_n(U)\};\qquad A_3\ :=\ \{\sigma_3 < M_n(U)\};
   \qquad A_4\ :=\ \{M_n(U) > m(n)\}.
\end{eqnarray*}
First, we note that ${\mathbb P}_{i-1}[A_0] \le 2n^{-3}$, from~\Ref{R-tilde-bnd},
for all~$n$ such that~\Ref{n-cond} is satisfied.  Then,
from Lemma~\ref{lema3}, for all $|i - nc| \le n\delta'_1$, as defined in~\Ref{U-def}, we have
\[
   {\mathbb P}_{i-1}[A_3 ]
      \ =\  {\mathbb P}_{i-1}\Bigl[\sup_{0\le u\le U} |n^{-1}Z_n(u) - c| > \delta/2 \Bigr]
   \ \le\  n^{-1}K_{U,\delta/2};
\]
then
\[
   {\mathbb P}_{i-1}[A_4 \cap A_3^c ] \ \le\  \exp\{-n\Lambda^*U/3\},
\]
by~\Ref{chernoff}; then, from~\Ref{sig1-def} and the definition of~$E_l$, and using~\Ref{L-ratio},
we have
\[
    A_1 \cap A_3^c \cap A_4^c \ \subset\ \bigcup_{l=1}^{m(n)} \left\{ \frac{L_1 E_l}{n\varepsilon} > 1 \right\} ,
\]
so that
\[
   {\mathbb P}_{i-1}[A_1 \cap A_3^c \cap A_4^c] \ \le\  m(n) e^{-n\varepsilon/L_1}.
\]
Finally, we immediately have
$
   A_2 \cap A_1^c \cap A_3^c \cap A_4^c \ \subset\ A_0
$
for all~$n \ge \max\{3,2\Lambda^*\}$ such that $n/\log n > 3(L_1/\varepsilon)^2$.
Combining these bounds with~\Ref{nearly}, and noting also that,
from Lemma~\ref{sdoi},
\[
   {\mathbb P}_{i-1}[Z_n(U) = k ] \ \le\  C_{\ref{sdoi}}(U)/\sqrt n,
\]
for all~$|i-nc|\le n\delta'_1$,  the lemma is proved.
\end{proof}

\begin{theorem}\label{supkpin}
Under Assumptions A1--A4, there exists a constant $C_{\ref{supkpin}}>0$ such that
$$
   \sup_{k\in \mathbb Z} |\Pi_{n}(k)-\Pi_{n}(k+1)|\ \le\  C_{\ref{supkpin}}\,n^{-1}\sqrt{\log n}.
$$
\end{theorem}

\begin{proof} Fix $U$ as in~\Ref{U-def}.
Since $\Pi_n$ is the equilibrium distribution of $Z_n$, it is in particular  true that
 \begin{eqnarray*}\nonumber
   \lefteqn{|\Pi_{n}(k)-\Pi_{n}(k+1)|
     \ =\  |\sum_{i \in \mathbb Z}  \Pi_n(i){\mathbb P}_i(Z_{n}(U)=k )
       -\sum_{i \in \mathbb Z}  \Pi_n(i){\mathbb P}_{i}(Z_{n}(U)=k+1 )|}\\\nonumber
    &&\leq \ \sum_{i \in \mathbb Z}  \Pi_n(i-1)
      |{\mathbb P}_{i-1}(Z_{n}(U)=k )-{\mathbb P}_i(Z_{n}(U)=k+1)|\\
    &&\qquad\mbox{} +\sum_{i \in \mathbb Z} |\Pi_n(i-1)- \Pi_n(i)|\,
                         {\mathbb P}_i(Z_{n}(U)=k+1).\phantom{HHHHHHHHHHHH}
\end{eqnarray*}
With $\delta'_1$ as in~\Ref{U-def}, note that one can write
 {\allowdisplaybreaks
\begin{eqnarray*}\nonumber
   \lefteqn{\sum_{i \in \mathbb Z}  \Pi_n(i-1)\,|{\mathbb P}_{i-1}(Z_{n}(U)=k)
      -{\mathbb P}_i(Z_{n}(U)=k+1)|}\\
   &&\leq\ \Pi_n\Big\{|Z_n+1-nc|> n\delta'_1\Big\}
      +\sup_{|i-nc|\le n\delta'_1} |{\mathbb P}_{i-1}(Z_{n}(U)=k)
          -{\mathbb P}_i(Z_{n}(U)=k+1)|,
\end{eqnarray*} }\\
 and that
\begin{eqnarray*}\nonumber
 \lefteqn{\sum_{i \in \mathbb Z} |\Pi_n(i-1)- \Pi_n(i)|{\mathbb P}_i(Z_{n}(U)=k+1)}\\
   &&\le\ \Pi_n\Big\{|Z_n+1-nc| > n\delta'_1\Big\}+\Pi_n\Big\{|Z_n-nc| > n\delta'_1\Big\}\\
   &&\quad\mbox{} +\sup_{|i-nc|\le n\delta'_1} {\mathbb P}_i(Z_{n}(U)=k+1) \cdot 2
       d_{TV}\{\Pi_n,\Pi_n*\delta_1\}.
\end{eqnarray*}
By applying the result of Corollary~\ref{prob-n1} three times, we obtain that
{\allowdisplaybreaks
\begin{eqnarray}\nonumber
   \lefteqn{\sup_{k\in \mathbb Z}|\Pi_{n}(k)-\Pi_{n}(k+1)|}\\\nonumber
   &&\ \le\   O(n^{-1})
     + \sup_{k\in \mathbb Z}\sup_{|i-nc|\le n\delta'_1} |{\mathbb P}_{i-1}(Z_{n}(U)=k)
      -{\mathbb P}_i(Z_{n}(U)=k+1)|\\\nonumber
   &&\qquad +\sup_{k\in \mathbb Z}\sup_{|i-nc|\le n\delta'_1} {\mathbb P}_i(Z_{n}(U)=k+1 )
     \cdot 2 d_{TV}\{\Pi_n,\Pi_n*\delta_1\}\\
   &&=:\ O(n^{-1})+\eta_{1n}+\eta_{2n}. \label{pk-pk1}
\end{eqnarray} }
The quantity $\eta_{1n}$ is of order $O(n^{-1}\sqrt{\log n})$, in view of Lemma~\ref{sunu};
and Lemma~\ref{sdoi} and  Theorem~\ref{thdtvpi} together give the bound
\begin{eqnarray}\nonumber
  \eta_{2n}
     &\le&  C_{\ref{sdoi}}(U)n^{-1/2}\cdot C_{\ref{thdtvpi}}\, n^{-1/2} \ =\  O(n^{-1}).
		 \label{bonds2}
\end{eqnarray}
This completes the proof of the theorem.
\end{proof}

 \setcounter{equation}{0}
 \setcounter{theorem}{0}
\section[Local limit theorem]{Local limit approximation for the equilibrium distribution}
The proof of Theorem~\ref{limth} consists of bounding
the quantities $R_{n1}$, $R_{n2}$ and~$R_{n3}$ of~\Ref{ek}, which all involve the
functions~$\tg_{nv_c,r}$ defined in~\Ref{defgt}.  For use in the subsequent argument, we collect
some of their properties.  We write $\|f\|_\infty := \sup_{i\in{\mathbb Z}}|f(i)|$,
$\|f\|_1 := \sum_{i\in\integ} |f(i)|$.

\begin{lemma}\label{2.4}
We have the following estimates:
\begin{eqnarray*}
  \brak1.&& \|\tg_{nv_c,r}\|_\infty \ \le\  \|\Delta\tg_{nv_c,r}\|_\infty \ \le\  1/(nv_c);\\
  \brak2.&& \|\Delta \tg_{nv_c,r}(i)\|_1 \ \le\  2/(nv_c); \\
  \brak3.&& \|\Delta^2 \tg_{nv_c,r}(i)\|_1 \ \le\  4/(nv_c); \\
  \brak4.&& |(i-\langle nv_c\rangle) \tg_{nv_c,r}(i)| \ \le\  h(i) + {\rm Po\,}(nv_c)\{r+{\lfloor nv_c \rfloor}\}; \\
  \brak5.&& |(i - \langle nv_c\rangle) \Delta\tg_{nv_c,r}(i)| \ \le\  h(i+1) + h(i) + 1/(nv_c),
\end{eqnarray*}
where, in parts 4 and~5, we have $h(i) \ge 0$ for all~$i$, and $\|h(i)\|_1 \le 3$.
\end{lemma}

\begin{proof}
For $i \le -{\lfloor nv_c \rfloor}$, $\tg_{nv_c,r}(i) = 0$;  for $i > -{\lfloor nv_c \rfloor}$, we have
$\tg_{nv_c,r}(i) = g_{\mu,s}(j)$, where $j = i+{\lfloor nv_c \rfloor}$, $\mu = nv_c$ and $s = r+{\lfloor nv_c \rfloor}$,
and $g = g_{\mu,s}$ satisfies the Stein--Chen equation
\begin{equation}\label{St-Poi}
   \mu g(j+1) - jg(j) \ =\  {\mathbf 1}_{\{s\}}(j) - {\rm Po\,}(\mu)\{s\},\qquad j \ge 0.
\end{equation}
Parts 1 and~2 now follow from the proof of Lemma~1.1.1 of
Barbour, Holst \& Janson~(1992), in which it is shown that the function
$g_{\mu,s}$ is negative and strictly decreasing in $\{1,2,\ldots,s\}$ and positive
and strictly decreasing in $\{s+1,s+2,\ldots\}$, with $\Delta g_{\mu,s}(s) \ \le\  1/(nv_c)$.
Part~3 is then immediate from part~2.

For part~4, using the notation above and~\Ref{St-Poi}, we have
\begin{eqnarray}
   \lefteqn{(i - \langle nv_c\rangle) \tg_{nv_c,r}(i) \ =\  (j-\mu)g_{\mu,s}(j)} \nonumber\\
   &&\ =\  \mu (g_{\mu,s}(j+1) - g_{\mu,s}(j)) - {\mathbf 1}_{\{s\}}(j) + {\rm Po\,}(\mu)\{s\}. \label{i-factor}
\end{eqnarray}
This implies that
\[
  |(i - \langle nv_c\rangle) \tg_{nv_c,r}(i)| \ \le\  \{\mu|\Delta g(j)| + {\mathbf 1}_{\{s\}}(j)\} + {\rm Po\,}(\mu)\{s\},
\]
which, with part~2, proves part~4.  It also follows immediately from \Ref{i-factor} that
\begin{eqnarray}
  |(i - \langle nv_c\rangle) \Delta\tg_{nv_c,r}(i)| \ \le\  h(i+1) + h(i) + |\tg_{nv_c,r}(i+1)|,
\end{eqnarray}
for the same function~$h(i) := \{\mu|\Delta g(j)| + {\mathbf 1}_{\{s\}}(j)\}$, and part~5 follows
on applying part~1.
\end{proof}

As a result of these bounds, combined with Theorems~\ref{thdtvpi} and~\ref{supkpin},
we can establish two useful bounds on expectations of differences of the~$\tg_{nv_c,r}(Y_n+\cdot)$,
under the equilibrium distribution.

\begin{lemma}\label{new-one}
For any $r,l\in{\mathbb Z}$, we have
\begin{eqnarray*}
  \brak1.&& {\mathbb E}|\bigtriangledown\tg_{nv_c,r}(Y_n+l)| \ \le\  \frac{2C_{\ref{thdtvpi}}}{n^{3/2}v_c};\\
  \brak2.&& |{\mathbb E}\{\bigtriangledown^2\tg_{nv_c,r}(Y_n+l)\}| \ \le\  \frac{2 C_{\ref{supkpin}}}{n^2v_c}\,\sqrt{\log n}.
\end{eqnarray*}
\end{lemma}

\begin{proof}
For the first part, it is immediate that
\[
    {\mathbb E}|\bigtriangledown\tg_{nv_c,r}(Y_n+l)| \ \le\  \sup_{i'\in{\mathbb Z}}\Pi_n(i')\,\sum_{i\in\integ}|\bigtriangledown\tg_{nv_c,r}(i)|.
\]
By Lemma~\ref{2.4}\,(2) and~\Ref{point-bnd}, this is bounded in turn by
\[
    d_{TV}\{\Pi_n,\Pi_n*\delta_1\}\,2/(nv_c),
\]
and part~1 follows from Theorem~\ref{thdtvpi}.  For the second part,
\begin{eqnarray}\label{term2insupk} \nonumber
  \lefteqn{|{\mathbb E}\{ \bigtriangledown^2 \tg_{nv_c,r}(Y_{n}+l)\}|} \\ \nonumber
  &&= \  \Bigl|\sum_{i\in \mathbb Z} \bigtriangledown \tg_{nv_c,r}(i-\lfloor nc\rfloor+s)
                (\Pi_n(i+1)-\Pi_n(i))\Bigr|\\
  && \leq\ \left\{ \sup_{i'\in \mathbb Z} \big|\Pi_{n}(i'-1)-\Pi_{n}(i')\big|\right\}\,
      \sum_{i\in \mathbb Z} |\bigtriangledown \tg_{nv_c,r}(i-\lfloor nc\rfloor)|\\\nonumber
  && \leq\  \sup_{i\in \mathbb Z} \big|\Pi_{n}(i-1)-\Pi_{n}(i)\big|\cdot 2(n v_c)^{-1},
\end{eqnarray}
where the last line uses Lemma~\ref{2.4}\,(2).  Part~2 of the lemma now follows from
Theorem~\ref{supkpin}.
\end{proof}

Bounding a further set of expectations that appear repeatedly in the estimates
first needs another, technical lemma.

\begin{lemma}\label{Lem2.5}
Let~$\mu$ be any probability distribution on~${\mathbb Z}$.  Suppose that $s,f$ and~$h$ are real
functions on~${\mathbb Z}$ such that $\|f\|_\infty < \infty$,  $\|\Delta s\|_\infty < \infty$
and $\|h\|_1 < \infty$, which also satisfy the inequality
\begin{equation}\label{sf-ineq}
  |s(i)f(i)| \ \le\  |h(i)| + k, \qquad I_1 \le i < I_2,
\end{equation}
 for some integers $I_1 < I_2$ and for some $k < \infty$.  Then
\begin{eqnarray*}
  \Bigl|\sum_{i=I_1}^{I_2} \mu_i s(i)\bigtriangledown f(i)\Bigr| &\le& \|f\|_{(I_1,I_2)}\,\|\Delta s\|_{(I_1,I_2)}
     + \|h\|_1 \sup_{I_1\le i < I_2}|\mu_i-\mu_{i+1}| + kd_{TV}(\mu,\mu*\delta_1)\\
   &&\mbox{}\qquad  + |\mu_{I_1}s(I_1)f(I_1-1)| + |\mu_{I_2}s(I_2)f(I_2)|,
\end{eqnarray*}
where $\|g\|_{(I_1,I_2)} := \sup_{I_1\le i < I_2}|g(i)|$.
\end{lemma}

\begin{proof}
It is immediate that
\begin{eqnarray*}
  \lefteqn{\Bigl|\sum_{i=I_1}^{I_2} \mu_i s(i)\bigtriangledown f(i)\Bigr|}\\
  && \ \le\
       \Bigl|\sum_{i=I_1}^{I_2-1}\{\mu_{i+1}s(i+1)-\mu_is(i)\}f(i)\Bigr| + |\mu_{I_1}s(I_1)f(I_1-1)|
           + |\mu_{I_2}s(I_2)f(I_2)|\\
  &&\ \le\  \Bigl|\sum_{i=I_1}^{I_2-1}\{\mu_{i+1}-\mu_i\}s(i)f(i)\Bigr|
      + \Bigl|\sum_{i=I_1}^{I_2-1}\mu_{i+1}\{s(i+1)-s(i)\}f(i)\Bigr|\\
   &&\mbox{}\qquad\qquad\qquad + |\mu_{I_1}s(I_1)f(I_1-1)|
           + |\mu_{I_2}s(I_2)f(I_2)|.
\end{eqnarray*}
Clearly, the second term is bounded by $\|f\|_{(I_1,I_2)}\,\|\Delta s\|_{(I_1,I_2)}$.  For the
first term, in view of~\Ref{sf-ineq}, we have at most
\[
  \sum_{i=I_1}^{I_2-1} \{|\mu_{i+1}-\mu_i|\,|h(i)|\} +  k\sum_{i=I_1}^{I_2-1}|\mu_{i+1}-\mu_i|,
\]
which is easily bounded by $\|h\|_1 \sup_{I_1 \le i < I_2}|\mu_i-\mu_{i+1}| + kd_{TV}(\mu,\mu*\delta_1)$,
in view of~\Ref{point-bnd}.

Note that the argument also goes through for $I_1=-\infty$ and $I_2=\infty$, in which
case the final two elements in the bound disappear.
\end{proof}

This lemma is combined with Lemma~\ref{2.4}\,(4) and~(5) to give the next
corollary, which is used as an ingredient in many of the estimates to be made.

\begin{corollary}\label{sf-appns}
Suppose that $|s(i)| \le |i - {\lfloor nc \rfloor}|$ for all~$|i| \le n\delta$.  Then, for any $0 < \delta'\le \delta$ and
all $l\in{\mathbb Z}$ such that $|l| \le n(\delta-\delta')$, we have
\begin{eqnarray*}
  \lefteqn{1.\quad |{\mathbb E}\{s(Y_n+l)\bigtriangledown\tg_{nv_c,r}(Y_n+l)I[|Y_n| \le n\delta']\}| }\\
   &&\ \le\  \frac1{nv_c}\,\sup_{|i|\le n\delta}|\Delta s(i)|
      + \frac{3C_{\ref{sunu}}}n \sqrt{\log n} + \frac{C_{\ref{thdtvpi}}}{2n\sqrt{v_c}}
      + 2(C_{\{\ref{2ineq},1\}} + C_{\{\ref{2ineq},2\}}/\delta')/(nv_c);\\
  \lefteqn{2.\quad |{\mathbb E}\{s(Y_n+l)\bigtriangledown^2\tg_{nv_c,r}(Y_n+l)I[|Y_n| \le n\delta']\}|}\\
   &&  \ \le\  \frac2{nv_c}\,\sup_{|i|\le n\delta}|\Delta s(i)|
      + \frac{6C_{\ref{sunu}}}n \sqrt{\log n} + \frac{C_{\ref{thdtvpi}}}{n^{3/2}v_c}
      + 4(C_{\{\ref{2ineq},1\}} + C_{\{\ref{2ineq},2\}}/\delta')/(nv_c).
\end{eqnarray*}
\end{corollary}

\begin{proof}
We take $\Pi_n*\delta_{-l}$ for~$\mu$ and either $\tg_{nv_c,r}$ or $\bigtriangledown\tg_{nv_c,r}$ for~$f$ in Lemma~\ref{Lem2.5},
noting that parts 4 and~5 of Lemma~\ref{2.4} give the appropriate counterparts
of~\Ref{sf-ineq}.  The first three elements appearing in the bound given by Lemma~\ref{Lem2.5}
are in turn bounded by using Lemma~\ref{2.4}\,(1), Lemma~\ref{sunu} and Theorem~\ref{thdtvpi}.
The last two are bounded by Lemma~\ref{2.4}\,(1) and Theorem~\ref{exubj}.
\end{proof}

\medskip
\noindent{\it Proof of Theorem~\ref{limth}}

\medskip
We are now in a position to undertake the proof of
Theorem~\ref{limth}, for which we need to bound the terms $R_{1n}$, $R_{2n}$ and $R_{3n}$ in~\Ref{ek}
to order $O(n^{-(\alpha+1)/2}\sqrt{\log n})$. First, we show that $R_{3n}$ is as
small as $O(n^{-3/2})$. This is because, from Barbour \& Jensen~(1989, Remark to Lemma 2.1),
if $X\sim{\rm Po\,}(\mu)$, then
$$
  \sup_{k\in \mathbb Z}\; \mathbb P (X=k)\leq \frac{1}{2\sqrt{\mu}}.
$$
Hence, and from Corollary~\ref{prob-n1}, it follows easily that
\begin{equation}\label{R3-bnd}
  R_{3n} \ =\  \sup_{k\in \mathbb Z} \;\widehat{{\rm Po}}(n v_c)\{k\}
    \cdot{\mathbb P}(Y_n< -\lfloor n v_c \rfloor)
    \ =\  O\Big(\frac{1}{n\sqrt{n}}\Big).
\end{equation}
For the quantity~$R_{2n}$ in~\Ref{ek}, we just use Lemma~\ref{new-one}\,(2) to give
\begin{equation}\label{R2-bnd}
   R_{2n} \ :=\ nv_c \sup_{r\in{\mathbb Z}}|{\mathbb E}\{ \bigtriangledown^2 \tg_{nv_c,r}(Y_{n}+1)\}|
     \ \le\  2C_{\ref{supkpin}}\,n^{-1}\sqrt{\log n}.
\end{equation}
It thus remains to bound~$R_{1n}$.  To do so, we consider in turn
the expectations of the quantities appearing in \Ref{rt} and in \Ref{En1}--\Ref{En5}.

Beginning with the elements of ${\mathbb E} R(n,r;Y_n)$, we first have
\begin{equation}\label{Rnr-1}
  {\mathbb E}\left\{\frac{n}{2}\Bigl[{\sigma}^2\Bigl(\frac{Y_n+\lfloor nc \rfloor}{n}\Bigr)
     -\sigma^2(c)\Bigr]\bigtriangledown{\tg_{nv_c,r}(Y_n)}\right\},
\end{equation}
which is of the form considered in Corollary~\ref{sf-appns}\,(1), with $l=0$ and
\begin{eqnarray*}
   s(i) &:=&  \frac{n}{2}\Bigl[{\sigma}^2\Bigl(\frac{i+\lfloor nc \rfloor}{n}\Bigr)
     -\sigma^2(c)\Bigr].
\end{eqnarray*}
For $|i| \le n\delta/2$ and $n \ge 2/\delta$, we have
\[
   |s(i)| \ \le\  {\textstyle\frac12}\,|i-\langle nv_c\rangle|\,\|(\sigma^2)'\|_\delta \quad\mbox{and}
   \quad |s(i)-s(i-1)| \le {\textstyle\frac12}\,\|(\sigma^2)'\|_\delta,
\]
whereas, for $|i| > n\delta/2$, we have the simple bound
\[
   |s(i)| \ \le\  \frac{n}{2}\Bigl[\sigma^2(c) + \sum_{j\in\integ} j^2c_j(1 + n^{-1}|i|)\Bigr],
\]
using Assumption~A2. By Theorem~\ref{exubj} and Corollary~\ref{prob-n1}, it follows
that the latter element contributes at most $O(n^{-1})$ to $|{\mathbb E} R(n,r;Y_n)|$; for the
former, Corollary~\ref{sf-appns} gives a bound of order $O(n^{-1}\sqrt{\log n})$.

For the next term,
\[
   {\mathbb E} \left\{ n\Bigl[F\Bigl(\frac{Y_n+\lfloor nc \rfloor}{n}\Bigr)-F(c)
       -\frac{Y_n}{n}F^{\prime}(c)\Bigr]\tg_{nv_c,r}(Y_n)\right\},
\]
$|\tg_{nv_c,r}(Y_n)|$ is bounded by $1/(nv_c)$, using Lemma~\ref{2.4}\,(1).
The contribution from the part $|Y_n| \le n\delta$ is thus easily bounded by
\[
   \bigl[\|F''\|_\delta\, n^{-2}\,{\mathbb E} \{Y_n^2 I[|Y_n \le n\delta]\} + \|F'\|_\delta n^{-1}\bigr]/v_c,
\]
and ${\mathbb E} \{Y_n^2 I[|Y_n \le n\delta]\}= O(n)$
by Theorem~\ref{exubj}, so that the whole contribution is of order $O(n^{-1})$.
If $|Y_n| > n\delta$, Assumption~A2 and Theorem~\ref{exubj} guarantee a contribution
of the same order.  The third term immediately yields
\[
   {\mathbb E}|F^{\prime}(c)\langle nv_c \rangle \tg_{nv_c,r}(Y_n)| \ \le\  |F^{\prime}(c)|/(nv_c),
\]
again of order $O(n^{-1})$.
All of these elements are of order $O(n^{-1}\sqrt{\log n})$, at least as small as the order
$O(n^{-(1+\alpha)/2}\sqrt{\log n})$ stated in the theorem, and it thus
remains to bound $|{\mathbb E}\{E_{nl}(\tg_{nv_c,r},Y_n+{\lfloor nc \rfloor})\}|$ for $1\le l\le 7$.

For the term arising from~\Ref{En1}, we have
\[
   {\mathbb E}\Bigl\{\frac{n}{2}\Bigl[F\Bigl(\frac{Y_n+\lfloor nc \rfloor}{n}\Bigr)
     -F(c)\Bigr]\bigtriangledown\tg_{nv_c,r}(Y_n)\Bigr\},
\]
which is of the form considered in Corollary~\ref{sf-appns}\,(1), with $l=0$ and
\begin{eqnarray*}
   s(i) &:=&  \frac{n}{2}\Bigl[F\Bigl(\frac{i+\lfloor nc \rfloor}{n}\Bigr)
     -F(c)\Bigr],
\end{eqnarray*}
and can be treated very much as was~\Ref{Rnr-1}, yielding a bound of the
same order.  For that arising from~\Ref{En2},
\[
  {\mathbb E} \left\{\sum_{j=2}^{{\lfloor \sqrt n \rfloor}} \Bigl[\sum_{k=2}^{j}\binom{k}{2}\bigtriangledown^2
       \tg_{nv_c,r}(Y_n+j-k+1)\Bigr]\,n \lambda_j(c)\right\},
\]
we can use Lemma~\ref{new-one}\,(2)
to bound the expectations ${\mathbb E}\bigtriangledown^2\tg_{nv_c,r}(Y_n+j-k+1)$, giving a contribution of at most
\[
   \sum_{j=2}^{{\lfloor \sqrt n \rfloor}}\frac16\, j^3c_j\,n \frac{2C_{\ref{supkpin}}}{n^2v_c}\,\sqrt{\log n}
      \ \le\  \frac{C_{\ref{supkpin}}\,s_\alpha}{3v_c}\,n^{-(1+\alpha)/2}\sqrt{\log n},
\]
where we have also used Assumption~A2.

The next term is from~\Ref{En3}, and is more complicated.  For its summands, we write
\begin{eqnarray}
   \lefteqn{\Bigl[-\binom{j}{2}\bigtriangledown\tg_{nv_c,r}(Y_n)+\sum_{k=1}^{j-1} k \bigtriangledown\tg_{nv_c,r}(Y_n+j-k)\Bigr]\,
       n \left\{\lambda_j\Bigl(\frac{Y_n+\lfloor nc \rfloor}{n}\Bigr) - \lambda_j(c)\right\}}\nonumber \\
  && \ =\  -\binom{j}{2}\bigtriangledown\tg_{nv_c,r}(Y_n)\,n
      \left\{\lambda_j\Bigl(\frac{Y_n+\lfloor nc \rfloor}{n}\Bigr) - \lambda_j(c)\right\} \nonumber\\
  &&\qquad\mbox{} + \sum_{k=1}^{j-1} k \bigtriangledown\tg_{nv_c,r}(Y_n+j-k)\,n
      \left\{\lambda_j\Bigl(\frac{Y_n+\lfloor nc \rfloor+j-k}{n}\Bigr) - \lambda_j(c)\right\} \nonumber\\
  &&\qquad\mbox{} + \sum_{k=1}^{j-1} k \bigtriangledown\tg_{nv_c,r}(Y_n+j-k)\,n
      \left\{\lambda_j\Bigl(\frac{Y_n+\lfloor nc \rfloor}{n}\Bigr)
         - \lambda_j\Bigl(\frac{Y_n+\lfloor nc \rfloor+j-k}{n}\Bigr)\right\} \nonumber\\
  && \ =\  E_{n3}^{(1)}(Y_n,j) +  E_{n3}^{(2)}(Y_n,j) +  E_{n3}^{(3)}(Y_n,j), \label{En3l}
\end{eqnarray}
say.  The term $E_{n3}^{(1)}(Y_n,j)$ is of the form considered in Corollary~\ref{sf-appns}\,(1),
with $l=0$ and
\begin{eqnarray*}
   s(i) &:=&  - n\binom{j}{2}
      \left\{\lambda_j\Bigl(\frac{i+\lfloor nc \rfloor}{n}\Bigr) - \lambda_j(c)\right\}.
\end{eqnarray*}
For $|i| \le n\delta/2$,
\[
   |s(i)| \ \le\  \binom{j}{2}\,|i-\langle nv_c\rangle|\,\|\lambda_j'\|_\delta \quad\mbox{and}
   \quad |s(i)-s(i-1)| \ \le\  \binom{j}{2}\,\|\lambda_j'\|_\delta,
\]
whereas, for $|i| > n\delta/2$, we have the direct bound
\[
   |s(i)| \ \le\  nc_j\binom{j}{2} (2 + n^{-1}|i|),
\]
using Assumption~A2.  From Corollary~\ref{sf-appns} and Assumption~A4, the contribution
from the first part is of order
\begin{equation}\label{En31-bnd}
   O\Bigl( c_j\binom{j}{2} n^{-1}\sqrt{\log n} \Bigr);
\end{equation}
the second part is also at most of this order, in view of Theorem~\ref{exubj},
Corollary~\ref{prob-n1} and~Lemma~\ref{2.4}\,(2).  Adding over $j \le {\lfloor \sqrt n \rfloor}$,
this gives a total contribution to the quantity $|{\mathbb E}\{E_{n3}(\tg_{nv_c,r},Y_n+{\lfloor nc \rfloor})\}|$ of order
$O(n^{-1}\sqrt{\log n})$.

For $E_{n3}^{(2)}(Y_n,j)$, we now have a sum of terms of the form considered in
Corollary~\ref{sf-appns}\,(1), with $l=j-k$ and
\begin{eqnarray*}
   s(i) &:=& nk \left\{\lambda_j\Bigl(\frac{i+\lfloor nc \rfloor}{n}\Bigr) - \lambda_j(c)\right\}.
\end{eqnarray*}
Supposing~$n$ to be large enough that $\sqrt n \le n\delta/2$, we have
\[
   |s(i)| \ \le\  k\|\lambda_j'\|_\delta\,|i-\langle nv_c\rangle| \quad\mbox{and}
   \quad |s(i)-s(i-1)| \ \le\  k\|\lambda_j'\|_\delta
\]
for $|i| \le n\delta/2$, whereas, for $|i| > n\delta/2$, we have the bound
\[
   |s(i)| \ \le\  nkc_j (2 + n^{-1}|i|).
\]
Arguing very much as for~\Ref{En31-bnd}, it thus follows that the total contribution
to the quantity $|{\mathbb E}\{E_{n3}(\tg_{nv_c,r},Y_n+{\lfloor nc \rfloor})\}|$ is again of order $O(n^{-1}\sqrt{\log n})$.

Finally, for $E_{n3}^{(2)}(Y_n,j)$, we again have a sum of terms.  We first note that
\begin{eqnarray*}
   \Bigl|\lambda_j\Bigl(\frac{i+\lfloor nc \rfloor}{n}\Bigr)
         - \lambda_j\Bigl(\frac{i+\lfloor nc \rfloor+j-k}{n}\Bigr)\Bigr|
     &\le& n^{-1}|j-k|\|\lambda_j'\|_\delta
\end{eqnarray*}
for $|i| \le n\delta/2$, and this leads to a contribution to~$|{\mathbb E} E_{n3}^{(2)}(Y_n,j)|$ of
at most
\begin{equation}\label{En33-1}
   \sum_{k=1}^{j-1} k(j-k)\|\lambda_j'\|_\delta/(nv_c) \ \le\  L_1j^3c_j/(6nv_c),
\end{equation}
in view of Lemma~\ref{2.4}\,(1).  For $|i| > n\delta/2$, there is the bound
\[
   |E_{n3}^{(2)}(i,j)| \ \le\  \sum_{k=1}^{j-1} \frac k{v_c}c_j\{2 + n^{-1}(2|i| + j-k)\},
\]
giving
\begin{equation}\label{En33-2}
   |{\mathbb E}\{E_{n3}^{(2)}(Y_n,j) I[|Y_n| > n\delta/2]\}| \ \le\  j^2c_j\{{\textstyle\frac76}\, {\mathbb P}[|Y_n| > n\delta/2]
      + n^{-1}{\mathbb E}(|Y_n|I[|Y_n| > n\delta/2])\},
\end{equation}
because $j \le {\lfloor \sqrt n \rfloor}$.  Adding \Ref{En33-1} and~\Ref{En33-2} over $j \le {\lfloor \sqrt n \rfloor}$ gives
a total contribution to $|{\mathbb E} E_{n3}^{(2)}(Y_n,j)|$ of order $O(n^{-(1+\alpha)/2})$, because
of Assumption~A2.

The term from~\Ref{En4} is much easier.  For $|i| \le n\delta$, we have the bound
\[
   \lambda_j\Bigl(\frac{i+\lfloor nc \rfloor}{n}\Bigr) \ \le\  c_j(1 + \delta),
\]
by Assumption~A2,
and ${\mathbb E}|\bigtriangledown \tg_{nv_c,r}(Y_n+l)| \le 2(C_{\ref{thdtvpi}}/v_c) n^{-3/2}$ for any~$l$, by
Lemma~\ref{new-one}. Hence
\begin{eqnarray*}
  \lefteqn{{\mathbb E}\Bigl|\Bigl[-\binom{j}{2}\bigtriangledown\tg_{nv_c,r}(Y_n)+\sum_{k=1}^{j-1} k
       \bigtriangledown\tg_{nv_c,r}(Y_n+j-k)\Bigr]\,n \lambda_j\Bigl(\frac{Y_n+\lfloor nc \rfloor}{n}\Bigr)
          I[|Y_n| \le n\delta]\Bigr|}\\
  && \ \le\   j^2 c_j (1+\delta) \frac{2C_{\ref{thdtvpi}}}{v_c}\, n^{-1/2},
   \phantom{HHHHHHHHHHHHHHHHHHHHHHHH}
\end{eqnarray*}
and summing over $j \ge {\lceil \sqrt n \rceil}$ gives a total contribution to~\Ref{En4} of at most
\[
     s_\alpha \frac{2C_{\ref{thdtvpi}}}{v_c}\,(1+\delta) n^{-(1+\alpha)/2},
\]
in view of Assumption~A2.
For $|Y_n| > n\delta$, the $j$-contribution is bounded by
\[
    j^2 c_j \|\bigtriangledown\tg_{nv_c,r}\|_\infty {\mathbb E}\{(n + |Y_n|)I[|Y_n| \ge n\delta]\}
       \ \le\  2 j^2 c_j (C_{\{\ref{2ineq},1\}} + C_{\{\ref{2ineq},2\}})/(nv_c),
\]
in view of Lemma~\ref{2.4}\,(1) and Theorem~\ref{exubj}, and summing over $j \ge {\lceil \sqrt n \rceil}$ gives
a contribution of order $O(n^{-1-\alpha/2})$.  Hence the complete contribution from~\Ref{En4}
is of order $O(n^{-(1+\alpha)/2})$.

The remaining terms \Ref{En5}--\Ref{En7} are treated in exactly the same way as those
in \Ref{En2}--\Ref{En4}.  In all, the largest order of any of the terms in \Ref{En1}--\Ref{En7}
is of order $O(n^{-(1+\alpha)/2}\sqrt{\log n})$, and since the other terms were of order $O(n^{-1}\sqrt{\log n})$,
Theorem~\ref{limth} is proved. \hfill$\qed$

 \setcounter{equation}{0}
 \setcounter{theorem}{0}
\setcounter{section}{4}
\section*{Appendix} \label{appendix}
The following results from~[SB] are used in the proofs.

\begin{theorem}{\rm [SB, Theorem 2.1.]}
\label{exubj} Under Assumptions A1--A4, for all~$n$  large enough, the process $Z_n$ has an
   equilibrium distribution $\Pi_n$, and
\begin{equation} \label{2ineq}
  \begin{split}
    &{\mathbb E}\{|n^{-1}Z_n-c| \cdot 1 \hspace*{-1mm} {\rm l}(|n^{-1}Z_n-c| > \delta)\}
              \ \le\ C_{\{\ref{2ineq},1\}}n^{-1};\\
    &{\mathbb E}\{(n^{-1}Z_n-c)^2 \cdot 1 \hspace*{-1mm} {\rm l}(|n^{-1}Z_n-c|\le \delta)\}
              \ \le\ C_{\{\ref{2ineq},2\}}n^{-1},
  \end{split}
\end{equation}
for $\delta$ as in Assumption~A3 and constants $C_{\{\ref{2ineq},1\}}$ and~$C_{\{\ref{2ineq},2\}}$;
as before, in such expressions, $Z_n$ is used to denote a random variable having the
equilibrium distribution~$\Pi_n$.
\end{theorem}

\begin{corollary} {\rm [SB, Corollary 2.5.]}  \label{prob-n1}
Under Assumptions A1--A4, for any fixed $\delta'$ such that $0 < \delta' \le \delta$,
there exists $C_{\ref{prob-n1}}(\delta') < \infty$ such that
$$
   {\mathbb P}[|n^{-1}Z_n-c|> \delta'] \ \le\   C_{\ref{prob-n1}}(\delta')n^{-1}.
$$
\end{corollary}

\begin{lemma}{\rm [SB, Lemma 3.1.]}\label{lema3}
Under Assumptions A1--A4, for any $U > 0$ and $0 < \eta \le \delta$, there exists a constant
$K_{U,\eta}<\infty$ such that
$$
   \mathbb P[\sup_{t\in [0,U]}|Z_n(t)-nc|> n\eta  \mid Z_n(0) = i]\ \leq\  n^{-1}K_{U,\eta},
$$
uniformly in $|i - nc| \le (n\eta/2) \exp\{-\|F'\|_{\delta}U\}$.
\end{lemma}

\begin{theorem} {\rm [SB, Theorem 3.2.]} \label{thdtvpi}
Under Assumptions A1--A4, there exists a constant $C_{\ref{thdtvpi}}>0$ such that
$$
   d_{TV}\{\Pi_{n},\Pi_{n}*\delta_1\} \ \le\  C_{\ref{thdtvpi}} n^{-1/2},
$$
where $\Pi_{n}*\delta_1$ denotes the unit translate of $\Pi_{n}$.
\end{theorem}

Finally, we shall use the following result, which was used in~[SB] to prove the
previous theorem; see, for example, (3.7) in the proof of [SB, Theorem 3.2].

\begin{lemma}\label{lema4}
Under Assumptions A1--A4,  for any $U\ge1$, there exists a constant $K_U<\infty$ such that
$$
   d_{TV}\{{\mathcal L}(Z_n(U) \,|\, Z_n(0)=i),{\mathcal L}(Z_n(U) \,|\, Z_n(0)=i)*\delta_1\} \ \le\  K_U n^{-1/2},
$$
uniformly in $|i - nc| \le n\delta/2$.
\end{lemma}

redpajama_set_name: RedPajamaArXiv


Satellite dishes sit on the roofs of homes in China's southwestern Sichuan province, Nov. 10, 2008.
Chinese authorities in the southwestern province of Sichuan are stepping up state television programs to Tibetan nomads ahead of a sensitive political anniversary next week and are making it harder for them to receive anything but state-controlled news, exile sources said.
Nomadic families in the Kardze (in Chinese, Ganzi) Tibetan Autonomous Prefecture were being given free television sets and satellite dishes, local officials said, confirming earlier reports.
"They are being given away free in the nomadic districts," said an official who answered the phone at the Xinlong county religious affairs bureau in Kardze on Tuesday.
Asked if the televisions contained surveillance equipment, the official said, "I don't know."
The Norway-based Voice of Tibet radio station reported on Monday that the television sets contained microphones which were part of a new surveillance drive ahead of next week's 60th anniversary of the "liberation" of Tibet by People's Liberation Army troops.
But while RFA was unable to confirm that report independently, exile sources and officials said Beijing was definitely extending the reach of state-backed news and programming into the region.
"Right now, what the Chinese government fears most is that Tibetans inside China will get information from outside the country," said Dharamsala-based Tibetan exile parliamentarian Kalsang.
"Obviously that makes life difficult for local Tibetans, so they have come up with this idea to install a television in their homes."
He said he believes reports that the televisions contain surveillance devices to be "probable."
Tsoge, a Chinese-language spokesman for the International Campaign For Tibet (ICT) based in Dharamsala, said he believes the report is at least partly credible.
"They probably would try that," he said. "For example, there are a lot of Tibetan officials that the government itself doesn't entirely trust."
"They would be very likely to put them under surveillance [at least]."
He said the authorities are also concerned about the number of Tibetans who listen to overseas broadcasts.
"In [Gansu's] Xiahe county there are a lot of people who listen to the Voice of America and Radio Free Asia," Tsoge said. "The nomads would listen to them on shortwave radio sets."
"They unlocked the sets' code and retuned them ... but after that the government took their receivers away."
The Sichuan authorities have already rolled out the government's own satellite television programming across the region in the Kham Tibetan dialect, official media reported earlier this month.
"Authorities in Sichuan provided the solar powered TV sets, worth 4,000 yuan (U.S. $620) each, to 20,000 poor farmers and herdsmen in the Ngaba and Kardze prefectures," Xinhua news agency reported on May 1.
The Kham dialect Kangba Satellite TV began broadcasting programs 18 hours a day in June 2010 to 2.4 million people living in Sichuan's Kardze and Ngaba, Khamdo in Tibet, Diqing in Yunnan, Yushu in Qinghai and Gannan in Gansu.
Lithang, one of the areas targeted by the television gifting program, is populated largely by Tibet nomads and has in the past been a flashpoint for protests in opposition to Chinese rule.
Ngaba was in recent weeks the scene of a harsh Chinese security crackdown following the self-immolation death of a monk from Kirti monastery protesting Beijing's rule.
China occupied Tibet in 1950 and claims the region has been part of its territory for centuries, although many Tibetans, who are linguistically and ethnically distinct, say they were effectively independent before that.
Monday's anniversary will mark the 60th anniversary of the signing of an accord which asserted Chinese sovereignty over Tibet and established a permanent military presence in the Himalayan region.
Reported by Rigdhen Dolma for RFA's Tibetan service, and by Qiao Long for the Mandarin service. Translated and written in English by Luisetta Mudie.


text:
Bernardino Pérez Elizarán, plus connu sous le nom de Pasieguito, né le  à Hernani (Guipuscoa) au Pays basque, et mort le  à Valence, est un joueur de football espagnol, qui évoluait au poste de milieu de terrain, avant de devenir entraîneur.

Il fait l'essentiel de sa carrière au Valence CF et compte trois sélections en équipe d'Espagne.

Devenu entraîneur, il dirige notamment le Valence CF et le CE Sadabell.

Biographie

Joueur
En tant que joueur, en 18 ans de carrière, Pasieguito a joué dans six clubs, dont le Valence CF (avec qui il remporte La Liga en 1943-44 et 1946-47 ainsi que la Copa del Rey en 1948-49 et 1954).

Entraîneur
En 22 ans de carrière d'entraîneur, Pérez Elizarán a en tout entraîné quatre clubs, dont Valence CF durant trois périodes différentes. Avec les Valenciens, il remporte la Supercoupe de l'UEFA en 1980.

Palmarès
Pasieguito remporte avec le Valence CF le Champion d'Espagne en 1944 et 1947, il termine vice-champion en 1949. Il gagne également deux Coupe d'Espagne en 1949 et 1954 avec Valence CF et, est finaliste de la compétition en 1944 et 1946.

Comme entraîneur, il amène le Valence CF à la victoire en Coupe d'Espagne en 1979 et en Supercoupe de l'UEFA en 1980. Sous ses ordres, le club est également finaliste de la Coupe des villes de foires en 1963-64.

Liens externes

  Pasieguito sur futbol.sportec.es
  Pasieguito sur www.ciberche.net

Notes et références

Naissance à Hernani
Naissance en mai 1925
Décès en octobre 2002
Footballeur international espagnol
Joueur du Valence CF
Joueur du Levante UD
Joueur du Burgos CF
Entraîneur espagnol de football
Entraîneur du Valence CF
Entraîneur du Grenade CF
Entraîneur du Real Sporting de Gijón
Décès à 77 ans

redpajama_set_name: RedPajamaWikipedia


text:
Robotic Kiosks: "20,000 Pizza Vending Machines Are Coming to North America"
by Administrator | Dec 29, 2021 | Kiosk, Technology, Vending
Photo: PizzaForno via Thrillist
"For years there's been buzz about a robot revolution, and it looks like it's here. PizzaForno Partners Les Tomlin and Will Moyer are leaning into the future and appeasing both the robot overlords and people who like good pizza fast by rolling out fully automated pizza kiosks across North America.
QSR Magazine reported that the pair plans to place 20,000 fully automated PizzaForno kiosks across the country by 2026. That number includes 1,000 kiosks in the United States by the end of 2022, with locations between Southern California, Louisiana, and Florida first.
'I think North America has been very late to the game on robotic food,' Tomlin told the outlet. 'I mean, look at PizzaForna—the technology has been around France for the better part of 10 years. COVID, the labor shortage, people don't want to spend 10 minutes waiting for anything. All those things add up to super fast, super convenient, super quick serve. That's where I think everybody's got to go.'
Customers who step up to a PizzaForno kiosk will be met with a 32-inch touch screen on which they can choose their pizza and how they plan to pay. The machine builds each pizza, and then a robotic arm removes it from the cold section, opens the lid, and puts it into a proprietary convection oven where it bakes. That takes between 90 and 120 seconds. The pizza is then dropped into a slot where customers can reach it. The entire process takes about three minutes. Customers can also take their pizza home cold and cook it themselves…"
— Caitlyn Hitt, Thrillist


text:
Sun Signs(current)
Birthday Horoscopes
The Most Accurate Enneagram Personality Test
Saul Williams Biography, Life, Interesting Facts
Birthday :
Also Known For :
Singer Musician Poet
Birth Place :
Newburgh, New York, United States of America
Zodiac Sign :
Chinese Zodiac :
Birth Element :
Rap artist Saul Williams has been making music. At the same time, he has used his talents as a poet, actor, and author.
Saul Williams was born on February 29, 1972, in Newburgh, New York. His father was a Baptist preacher. His mother was a teacher.
Saul Williams graduated from Newburgh Free Academy. He attended Morehouse College where he earned his degree in acting and philosophy. Williams graduated from New York University's Tisch School of Arts with a Master of Fine Arts degree.
Do You Have Good Karma?
Try The Karma Quiz Now!!
Rise To Stardom
Saul Williams got his start to fame for his poetry. In 1996, he won the "Grand Slam Championship" at the Nuyorican Poets Cafe.
His acting, however, put him into a larger spotlight. While he had a few roles in some films, it was when he was the lead actor in the 1998 film Slam.
At the same time that Slam was released, Saul Williams was also working as an artist with others such as Christian Alvarez, Erykah Badu, Blackalicious, and Sonia Sanchez of the Black Arts Movement. Saul Williams released his LP in 2001. Amethyst Rock Star was his first studio album in the genre of experimental hip-hop. It received mixed reviews from critics. He released Saul Williams in 2004 with greater success. Saul Williams toured with other groups and reached a broader audience. He went on the 2005 European tour with Nine Inch Nails and performed at Lollapalooza. He also collaborated with Nine Inch Nail's Trent Reznor on his 2007 album The Inevitable Rise and Liberation of NiggyTardust!.
At first, the album was only available via the Internet as a digital download. Willilams appeared in Nine Inch Nail's Year Zero album that criticized contemporary U.S. Politics. Along with releasing albums, touring the world, and acting in various television, film, and theater roles, Williams continues to write and publish poetry. The vocal reading of NGH WHT – The Dead Emcee Scrolls with the Arditti Quartet was released in 2009. The Arditti String Quartet conducted by Thomas Kessler is also on the album.
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Personal Life and Legacy
Saul Williams had a relationship with artist Marcia Jones. They had a daughter. Williams married Persia White in 2008. The couple divorced after a year together.
Philanthropic Works / Humanitarian Work
Saul Williams is a critic of the War on Terrorism, in particular, the wars in Iraq and Afghanistan. He also signed the Occupy Movement to take a stand against social and economic inequality.
For his portrayal of a young poet, he was nominated for the Best Debut Performance for the Independent Spirit Award and won the Breakthrough Award from Gotham Awards.
Summary of Major Works
Slam (1998), King of the Korner (2000), Today (2012)
Holler if Ya Hear Me (2014)
Girlfriends (2003), Lackawanna Blues TV movie (2005)
Saul Williams (2004), The Inevitable Rise and Liberation of NiggyTardust! (2007), NGH WHT – The Dead Emcee Scrolls with The Arditti Quartet (2009), Volcanic Sunlight (2011)
Poetry Collections
The Seventh Octave (1998), She (1999), Said the Shotgun to the Head (2003), The Dead Emcee Scrolls (2006)' Chorus (2012)' US (2015)
February 29 Horoscope
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Edwin Mattison McMillan
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Brandon Tartikoff
Terry Bradshaw
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text:
Complications of Bariatric Surgery
Bariatric surgery encompasses a collective group of surgical procedures that involve altering the digestive tract in order to promote weight loss. In general, bariatric surgery is considered for people who are extremely obese, with a body mass index (BMI) of 40 or higher. People with a BMI of 35 to 39.9 may also be eligible for bariatric surgery if they also have serious weight-related comorbid conditions such as type 2 diabetes, hypertension, or sleep apnea syndrome. Typically, patients who are considering bariatric surgery as a weight los option will have to undergo an extensive screening process and also meet specific guidelines to qualify for bariatric surgery. According to the American Society for Metabolic and Bariatric surgery, the total number of metabolic and bariatric surgeries performed in the United States in 2018 was 252,000.
In general, there are 3 types of bariatric surgical procedures that have evolved over the past 3 decades:
1. Blocking procedures, such as biliopancreatic diversion, that aim to block the absorption of food from the stomach.
2. Restrictive procedures, such as sleeve gastrectomy, that shrink the size of the stomach so that the patient feels more satiated while eating less amounts of food.
3. Mixed procedures, the most common of which is Roux-en-Y gastric bypass, that employ a combination of both blocking and restrictive strategies simultaneously.
Weight loss surgery is one of the fastest growing segments of all surgical procedures. Although bariatric procedures are generally considered safe and effective, they can be associated with serious complications, some of which can be fatal if not recognized quickly. The most dreaded complication is an anastomotic leak which is associated with a significant increase in both morbidity and mortality. In fact, a study conducted in 2017 of 140 medicolegal claims in bariatric surgery found that the most common reason for a malpractice claim was a delay in diagnosis or treatment of a complication in the postoperative period, the most common of which was an anastomotic leak. Death was reported in 52% of these cases. Nearly 50% of the cases were decided in favor of the patient, with a median award payout of $1,090,000.
In addition to anastomotic leaks, patients who undergo bariatric surgery can develop a variety of other early or late postoperative complications. Bariatric surgeons must be vigilant to identify these complications as soon as possible and manage these potentially life-threatening complications promptly.
The MediFocus Literature Guide to Complications of Bariatric Surgery captures the clinically relevant articles published over the past 10-years in peer-reviewed medical journals and focuses on the risk factors, diagnosis, management, and prevention of these potentially serious and life-threatening complications. This unique Literature Guide consists of over 180 hand-picked articles with links to the Abstracts and includes online access to the FREE FULL-TEXT version of 88 articles. The Guide also includes a valuable Author Directory for quickly identifying and locating medical authorities for case evaluation and expert testimony.
Complications of Bariatric Surgery is a one-of-a-kind literature reference Guide that includes:
A unique "Author Directory" consisting of the names and institutional affiliations of experts who have published and have specialized knowledge about Complications of Bariatric Surgery. The "Author Directory" is a valuable resource for quickly identifying and locating experts for case reviews, opinions, and testimony.
Medical Malpractice in Bariatric Surgery: a Review of 140 Medicolegal Claims.
Management and Prevention of Surgical and Nutritional Complications After Bariatric Surgery.
Rate of Death and Complications in Laparoscopic and Open Roux-en-Y Gastric Bypass.
Medico-legal analysis of legal complaints in bariatric surgery: a 15-year retrospective study.
Complications of bariatric surgery–What the general surgeon needs to know.
American Society of Metabolic and Bariatric Surgery patient safety committee policy statement on the qualifications of expert witnesses in bariatric surgery medicolegal matters.
Bariatric surgery claims – a medico-legal perspective.
Bariatric surgery: malpractice risks and risk management guidelines.
Benefits and Risks of Bariatric Surgery in Adults: A Review.
Lymphatic leakage after laparoscopic sleeve gastrectomy; presentation, management and review of the literature.
Association of Maternal Gastric Bypass Surgery With Offspring Birth Defects.
Obesity Surgery and Anesthesiology Risks: a Review of Key Concepts and Related Physiology.
30-Day Outcomes After Intraoperative Leak Testing for Bariatric Surgery Patients.
A 23-year study of mortality and development of co-morbidities in patients with obesity undergoing bariatric surgery (laparoscopic gastric banding) in comparison with medical treatment of obesity.
Late postoperative bleeding after Roux-en-Y gastric bypass: management and review of literature.
Hypoglycemia After Gastric Bypass Surgery: Current Concepts and Controversies.
American Society for Metabolic and Bariatric Surgery review of the literature on oneanastomosis gastric bypass.
Preventing Wernicke Encephalopathy After Bariatric Surgery.
Association of Bariatric Surgery vs Medical Obesity Treatment With Long-term Medical Complications and Obesity-Related Comorbidities.
Endoscopic Evaluation and Treatment of Postoperative Bariatric Surgery Complications.
The MediFocus Literature Guide on Complications of Bariatric Surgery


Have supper or high tea by the fire in your Shepherd's Hut, freshly prepared here. Have a barbeque or open fire in your fire pit, right outside your hut and admire the view. Explore our 150 acres of patchwork fields down to the river in the valley below.
Fly a kite or play a ball game in our fields, or tennis on the court. Head into the indoor beach for a game badminton or wet weather sandcastle building. Make friends with our friendly animals, from ponies, horses, goats, sheep, rabbits, pig to guinea pigs. Laze with a book against the cushions on your super comfy bed in your shepherd's hut, or simply gaze through the window at that view.
Duloe Community Shop: our village shop is now run by the community and stocks a good range of provisions as well as excellent local veg, newspapers etc. Please support it! Open Monday – Friday 8am-5pm (shut for lunch 1-3) and Saturday 8am-1pm.
The Blue Plate: Delicious sea food and local meats and produce – food for any mood – at a great new restaurant run by renowned local chef Nick Barclay.
Rick Stein's Restaurants: There are four restaurants in Padstow where you can try his food, The Seafood Restaurant , St Petroc's Bistro , Rick Stein's Café and Stein's Fish & Chips . Plus a pub in St Merryn called The Cornish Arms.
We have a licensed riding school at Treworgey. We offer riding lessons and some supervised hacking through the beautiful countryside around Treworgey, and across 100 acres of our own rolling farmland down to the Looe river below us.
Following these suggestions, and simply staying at the shepherd's hut saves a significant amount of energy compared with a more traditional type of holiday.
Looe Shark Angling Club: The premier shark angling centre in Great Britain.
Hideaway Huts provides an ideal base for a walking holiday. There are so many beautiful walks in the district, including right from either hut and on the farm. You can even reach us by bus or train and then on foot and leave the car at home.
Walks from the Looe Valley Line: The following Looe Valley Line Trails are all accessible from and end at stations along the Looe Valley Line.


This is a really exciting opportunity for people who have experience of arthritis whether personal or via family and friends – AND for those willing to learn – to provide peer support to others living with arthritis.
For a few hours each week – or less regularly if required – you will be helping us to run drop-ins in the West Berkshire area: currently Central Reading, Pangbourne, Wokingham, Caversham and Newbury. You can choose your local area and we will be adding new areas, so please do get in touch.
You will be supported by Arthritis Care staff at every drop in and we will make sure you are confident and comfortable before you start in your role.
Arthritis Care is the UK's largest organisation working with and for all people with arthritis. We are a membership organisation with a network of local branches and groups throughout England, Scotland, Wales and Northern Ireland.
Our vision is a world where people with arthritis can lead full and active lives.
Our purpose is to empower people with arthritis through support and information, ensuring their voices are heard and their conditions more effectively managed.
If you're friendly, helpful and approachable then you are just what we need. If you enjoy getting out and about in your local community we would love to hear from you.
Ideally someone who has experience with arthritis whether personal or via family and friends – AND for those willing to learn – who would like to meet and provide support and information for others living with the condition.
Mon-Fri AM&PM: ANY of these times, to suit volunteer.
← Help make Cowley Road Carnival Happen!


If the quality and quantity of books published in a year are an indicator of healthy discourse about the current state of India's environmental movement, 2018 has been a very good year for Indian green literature. The mix weighs, as it should be, towards how to address India's seemingly unsurmountable problems, and interestingly, in uniquely Indian ways. They are written by the very best in business which ensures greater credibility, an important factor in the post truth world.
Although launched at the end of December 2017, Sunita Narain sets the tone for 2018 with her Conflict of Interest: My Journey Through India's Green Movement. It's a personal account of her battles in raising the level of debate and environmental consciousness among the policy makers, businesses and common citizens. Narain, unlike most authors, doesn't mince her words and goes for the jugular.
It's because of her and a handful of activists, public policy in India is not being easily hijacked by interest groups. Her NGO Centre for Science & Environment and Down to Earth magazine are by far the most aggressive and informative on India's struggle for balancing convenient political and business interests against that of the citizens and the environment.
The much delayed and anticipated Aruna Roy's book – The RTI Story: Power to the People – is a heart-wrenching story of people's struggle for their basic human rights. While her phenomenal story of influencing the Right to Information legislation is well known, the book is about how the movement unfolded which every Indian should read. It unfolds the drama of a long-fought grueling movement. Despite recent attempts to dilute, RTI is still the single most effective weapon in the hands of a citizen against brazen politicians and bureaucrats.
She has been one of the big influencers on the new bunch of leaders leading several recent movements by farmers and underprivileged caste and classes. Her book is a big motivator for young rebels to stay in the fight for social and economic justice despite harsh conditions.
Mridula Ramesh's The Climate Solution: India's Climate-Change Crisis and What We Can Do about Itis a book of hope as she writes stories of 'warriors' who are able to use acumen and business sense to change the world around them. She shows how climate warriors, from the cotton fields of Punjab and eco start-ups in Bengaluru, to a forest guardian in Assam and the Johads of Rajasthan, have employed ingenuity and initiative to adapt to the changing conditions – and sometimes reverse their shattering effects.
This book is an urgent call to action – and an essential manifesto for policy makers, investors and entrepreneurs. It should inspire graduate students to take up entrepreneurship as a calling and as the best way to make a positive impact.
Naina Lal Kidwai's Survive or Sink: An Action Agenda for Sanitation, Water, Pollution and Green Finance is another call-to-action book. She argues for a collaborative approach to solve India's massive problem of sanitation and pollution. Being a star banker, Naina delves into the world of green finance quite intricately and shows how finance is not a challenge for India if it's serious about mitigating climate induced challenges. She shows how, if done well, climate change is a big opportunity for massive employment and inclusive growth.
Prem Shankar Jha's Dawn of the Solar Age – An End to Global Warming and to Fear is one of the most convincing books I've read on climate change. He has made compelling argument in favour of India changing its fuel mix urgently and resorting to proven alternative fuels like methanol and ethanol. He shows how this policy, if implemented seriously, will have a wide positive impact on rural economies and sustainability of agri-business. This strategy not only addresses climate change problems but has the potential of turning India into a climate positive nation by 2030.
This Book is Not Rubbish: 50 Ways to Ditch Plastic, Reduce Rubbish and Save the World! By Isabel Thomas is a very timely book about how citizens can make a major difference to their lives and their surroundings by adopting simple tools.
The good news is that there are loads of easy ways one can make a difference without waiting for big projects or policies. Even while covering serious issues like plastics, pollution, global warming and endangered animals, this book is a nice guide to people who want to become eco-warriors, not eco-worriers. It challenges consumers to take control of their future and thereby help clear the world of all the rubbish.
Top scientist Raghunath Mashelkar's book Leap Frogging to Pole-Vaulting: Creating the Magic of Radical Yet Sustainable Transformation just made it to the 2018 list. This too is a 'to do' book. It tries to convince readers to make a shift from reactive leapfrogging to proactive pole-vaulting to solve what appears to be daunting problems.
Two books published in 2016 need mention for their sheer force of ideas. One is Prof. Anil Gupta's Grassroots Innovation: Minds On the Margin Are Not Marginal Minds. He forcefully and pragmatically shows how the economically poor are not poor of ideas. His Honey Bee Network brought together thousands of grassroots innovators who went on to become successful entrepreneurs. With examples like Mitti Cool Refrigerator to the Footbridge at Meghalaya he insists that to fight the largest and most persistent problems of the world we must eschew expensive research labs and instead look towards ordinary folk.
India was such a place until colonialism and socialism changed them into waiting for someone to solve their problems. It's time to go back. It's being realized that the model local solutions to global problems works better than the other way around.
The Great Derangement: Climate Change and the Unthinkable, published in 2016, by Amitav Ghosh, is perhaps the most provocative. He is appalled at the indifference and casualness of people to what's happening to the Earth's climate.
He partly blames fiction writers for the current mess. Ghosh examines the inability of literature, history and politics to grasp the scale and violence of climate change. In fiction, he writes, hundred-year storms and freakish tornadoes simply feel too improbable for the novel and are automatically consigned to other genres.
What's interesting is that all these books were published by prominent publishing houses. Goes to show that issues about climate, ecology and environment are finally coming into the mainstream. Hope these books do well and inspire more such work.


text:
Home Uncategorized "Jason Gould Night" Friday at Warrior Gymnasium
"Jason Gould Night" Friday at Warrior Gymnasium
Blake Branch
(Blake Branch/Press-Herald Photo) Former Lakeside head coach Jason Gould coaches while his son, Jalen Gould, looks on during a 2016 home game at Warrior Gymnasium.
Over the offseason, longtime Lakeside head basketball coach Jason Gould stepped down from his post, ending a marvelous 23-year run as head coach of the Warriors. Now, the school will recognize him for a job well done.
Lakeside hosts Green Oaks Friday night in Sibley, and in between the girls and boys games, the school will honor his career and accomplishments with a ceremony inside Warrior Gymnasium.
Gould's career began at Central School in Grand Cane. In the 1990-1991 season, the Pioneers went 3-21, but that experience led to an assistant coaching position at Minden High School where Gould stayed on for three seasons.
In 1994, Sibley High School, not yet Lakeside, came calling for Gould, and a beautiful relationship began. During that first season, Gould and the Warriors won 31 games, brought home a district title and made a run to the state semifinals.
Along the way, Lakeside would make 13 playoff appearances and appear in one more final four, winning 350 games under Gould's tutelage. His numbers are even more impressive for those who were present to witness the quality of basketball played during the late 1990's – early 2000's. Matchups in Warrior Gymnasium against Minden, Springhill, Evangel, Doyline and other thriving programs of the time were considered must-see.
Gould coached two players, T.J. Carpenter (Nicholls State) and Jermaine Wallace (Northwestern State), to Division I Universities. You may remember Wallace for hitting the game-winning shot in Northwestern State's thrilling upset of Iowa in the 2006 NCAA Tournament.
Gould is still hanging around the halls and gymnasium in Sibley, helping mentor the current head coach of the Warriors, Marlon McDaniel, who played under Gould himself from 2006-2009 and assisted Gould for the past two seasons.
McDaniel couldn't quit gushing over Gould's leadership and willingness to help anyone for the betterment of Lakeside's hoops program.
"Though it's been just under a decade since Coach Gould coached me, it seems like last year," McDaniel said. "I can remember him pushing us to go harder every day; He's still coaching me. Anytime I need advice or a different point of view he's there to give it. We talk often and I feel a little wiser after each time… It's really a blessing I don't take for granted. I don't know his shoe size, but it'll be tough filling them."