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Australian strategic policy expert welcomes potential for US Naval base
key enablers | 16 December 2019 | Stephen Kuper
By: Stephen Kuper
Ross Babbage, an Australian strategic policy specialist, has welcomed revelations of a $715 million expansion of naval infrastructure in the Northern Territory, providing an opportunity for further basing of major US Navy units as both nations seek to reorientate their forces to better respond to rising great power competition in the Indo-Pacific.
Since former US president Barrack Obama announced a reinvigorated US presence in the Indo-Pacific as part of the Pacific Pivot in 2013, Darwin has emerged as one of the key focal points for US strategic planners and the Australian Defence Force, as the nation responds to an increasingly assertive China and rapidly evolving economic, political and strategic environment.
Located in close proximity to the strategic sea lines of communication (SLOC) of Malacca, Sunda and Lombok, Darwin is also Australia's gateway to the Indo-Pacific, serving as a launching point for Australia's economic and strategic engagement with the region.
While the broader economic potential of Darwin is heavily under-utilised, the strategic potential of the city is equally under-utilised, particularly given the rise of Indo-Pacific Asia and China – something increasingly recognised by the US as it seeks to re-position itself in the region.
As the regional dynamics have changed, successive Australian governments have sought to re-position key Australian military assets throughout the northern approaches to the landmass.
However, it wasn't until the 2016 Defence White Paper that this tactical and strategic reorientation was set in stone, with the white paper identifying the need for Australia to shift beyond the narrow sea-air gap with the aforementioned facilities serving as key staging points for Australia's engagement in the region.
"Australia's strategic outlook to 2035 also includes a number of challenges which we need to prepare for. While there is no more than a remote prospect of a military attack by another country on Australian territory in the foreseeable future, our strategic planning is not limited to defending our borders," the white paper identified.
"Our planning recognises the regional and global nature of Australia's strategic interests and the different sets of challenges created by the behaviours of countries and non-state actors such as terrorists."
On the back of these factors, there is a growing consensus among Australia's strategic policy apparatus that the north of the continent is going to play an increasingly important role in the nation's future force posture and geo-strategic, economic and political engagement with the Indo-Pacific.
A key driving factor behind this shifting position is the growing need to provide greater support for the US Navy as it seeks to counter an increasingly assertive China in the Indo-Pacific and the US faces a major shortfall of secure military infrastructure in the region between the Persian Gulf and major US infrastructure in Japan or South Korea.
Increased Australian and allied military presence
Speaking to Seth Robson of Stars and Stripes, Australian strategic policy expert Ross Babbage recently discussed the $715 million worth of infrastructure modernisation, upgrades and expansion at key Defence locations in and around Darwin to demonstrate the collaborative relationship between the US and Australia.
The improvements will include a focus at HMAS Coonawarra and the Larrakeyah Defence Precinct, with the enhancements designed to support increased Australian and US military rotations and presence in the region – this is expected to include the Royal Australian Navy's future Arafura Class offshore patrol vessels and an increased US Navy presence building on the increased rotations.
This year saw an increased US military presence ranging from the largest Marine Air-Ground Task Force deployment, resulting in 2,500 US Marines deployed to the Marine Rotational Force-Darwin (MRF-D), and the guided-missile destroyer USS Stockdale, the mine countermeasures ship USS Patriot and the submarine tender USS Emory S. Land.
Recognising the increasing number of US forces rotating through Darwin, the $272 million infrastructure investment is designed to support an increasing presence of US and Australian major surface combatants in the port.
Building on this, key air bases surrounding Darwin, including RAAF Base Tindal, are also expected to receive $88.65 million worth of infrastructure modernisation including fuel storage, expanded airfield and maintenance facilities.
An opportunity for joint Australian, US and UK defence facilities and basing
While much has been made of an expanded US presence in Australia, particularly in northern Australia, centred on Darwin, the UK's commitment to expanding its regional presence, with a focus on south-east Asia, provides both Australia and the US with an opportunity to share the financial costs associated with redeveloping major defence infrastructure, while also further enhancing joint force interoperability.
Both Darwin and the Northern Territory frequently play host to multinational capability building exercises such as Exercise Diamond Storm, Exercise Southern Jackaroo, Exercise Hamel and Exercise Lightning Focus.
Reinforcing the strategic importance of the state and city in long-term strategic planning, rumours have recently been circulating about the potential of Darwin and the Northern Territory will play host to a larger US Marine Corps presence.
The dispersed nature of the Northern Territory defence infrastructure, combined with the large-scale basing requirements of forward-deployed US military assets, provides an opportunity to hit reset on key defence infrastructure – particularly accommodations, ship mooring and basic, and in some cases in-depth, maintenance and sustainment and airfield requirements – to develop a series of joint military facilities capable of supporting long-range, sustained combat operations throughout the Indo-Pacific.
An example of this could include the major redevelopment of naval facilities in Darwin to accommodate both Australian, British and American expeditionary strike groups, with specialised moorings to accommodate a US Navy Nimitz or Gerald R. Ford Class and the Royal Navy's new Queen Elizabeth Class supercarriers and supporting naval task group – providing an alternative basing arrangement to the comparatively vulnerable facilities existing in Japan and Guam.
Further supporting this is the growing platform commonality and resulting interoperability, beginning with the Aegis platforms operated by the US and Australian navies, the E-7A Wedgetail currently operated by the Royal Australian Air Force, the Republic of Korea Air Force and soon to be operated by the Royal Air Force, and of course the Lockheed Martin F-35 Joint Strike Fighter and Type 26 Global Combat Ships.
Babbage explained, "The details are still being sorted out, but you will see quite a bit more in the next three to four years."
The nation is defined by its relationship with the region, with access to the growing economies and to strategic sea lines of communication supporting over 90 per cent of global trade, a result of the cost-effective and reliable nature of sea transport.
Indo-Pacific Asia is at the epicentre of the global maritime trade, with about US$5 trillion worth of trade flowing through the South China Sea and the strategic waterways and chokepoints of south-east Asia annually.
Enhancing Australia's capacity to act as an independent power, incorporating great power-style strategic economic, diplomatic and military capability serves as a powerful symbol of Australia's sovereignty and evolving responsibilities in supporting and enhancing the security and prosperity of Indo-Pacific Asia.
Shifting the public discussion away from the default Australian position of "it is all a little too difficult, so let's not bother" will provide unprecedented economic, diplomatic, political and strategic opportunities for the nation.
However, maximising the effectiveness of this response to regional affairs requires a dramatically different approach to the nation's key defence infrastructure and staging points to support enhanced tactical and strategic response times and balance, taking into account the dramatically different operational environments.
Get involved with the discussion and let us know your thoughts on Australia's future role and position in the Indo-Pacific and how basing infrastructure will support the capability of the ADF as the regional balance of power continues to evolve, or get in touch with This email address is being protected from spambots. You need JavaScript enabled to view it. or at This email address is being protected from spambots. You need JavaScript enabled to view it.
Aussie SME provides 'eye in the sky' capability to support bushfire fight
Enhancing the value of northern Australia in the era of great power competition
Boeing welcomes next president for Australia, New Zealand and South Pacific operations
Stephen Kuper
Last Updated: 16 December 2019 Published: 16 December 2019
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text:
\section{Introduction}
\setcounter{equation}{0}

Consider a family $\mathbf X = \{X_{jk}\}$, $1 \leq j \leq k \leq n$,
of independent real random variables defined on some probability space
$(\Omega,{\textfrak M},\Pr)$, for any $n\ge 1$. Assume that $X_{jk} = X_{kj}$, for
$1 \leq k < j \leq n$, and introduce the symmetric matrices
\begin{displaymath}
 \mathbf W = \ \frac{1}{\sqrt{n}} \left(\begin{array}{cccc}
 X_{11} &  X_{12} & \cdots &  X_{1n} \\
 X_{21} & X_{22} & \cdots &  X_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
 X_{n1} &  X_{n2} & \cdots &  X_{nn} \\
\end{array}
\right).
\end{displaymath}

The matrix $\mathbf W$ has a random spectrum $\{\lambda_1,\dots,\lambda_n\}$ and an
associated spectral distribution function
$\mathcal F_{n}(x) = \frac{1}{n}\ {\rm card}\,\{j \leq n: \lambda_j \leq
x\}, \quad x \in \R$.
Averaging over the random values $X_{ij}(\omega)$, define the expected
(non-random) empirical distribution functions
$ F_{n}(x) = \E\,\mathcal F_{n}(x)$.
Let $G(x)$ denote the semi-circular distribution function with density
$g(x)=G'(x)=\frac1{2\pi}\sqrt{4-x^2}\mathbb I_{[-2,2]}(x)$, where $\mathbb I_{[a,b]}(x)$
denotes an indicator--function of interval $[a,b]$. We shall study the
rate of convergence of
 $\mathcal F_{n}(x)$ to the semi-circular law under the condition
\begin{equation}\label{exptails}
 \Pr\{|X_{jk}|>t\}\le \varkappa^{-1}\exp\{-t^{\varkappa}\},
\end{equation}
 for some $\varkappa>0$ and for any $t\ge1$.
 The rate of convergence to the semi-circular law has been studied by several authors.
 We proved  in \cite{GT:03}  that the  Kolmogorov distance between $\mathcal F_n(x)$ and
 the distribution function
 $G(x)$,
 $\Delta_n^*:=\sup_x|\mathcal F_n(x)-G(x)|$ is of order
 $O_P(n^{-\frac12})$ (i.e. $n^{\frac12}\Delta_n^*$ is bounded in probability). Bai \cite{Bai:02} and Girko  \cite{Girko:02}
 showed that  $\Delta_n:=\sup_x| F_n(x)-G(x)|=O(n^{-\frac12})$.
 Bobkov, G\"otze and Tikhomirov \cite{BGT:08}  proved that $\Delta_n$ and $\E\Delta_n^*$ have order $O(n^{-\frac23})$
assuming a Poincar\'e inequality for the
distribution of the  matrix elements. For the Gaussian Unitary Ensemble respectively for the Gaussian Orthogonal Ensemble, see  \cite{GT:05}
respectively  \cite{Tim:07},
it has been shown that $\Delta_n=O(n^{-1})$.
Denote by $\gamma_{n1}\le\ldots\le \gamma_{nn}$, the quantiles of $G$,
i.e.   $G(\gamma_{nj})=\frac jn$.
We introduce the notation
 $\operatorname{llog}_n :=\log\log n$.
Erd\"os, Yau and Yin \cite{ErdosYauYin:2010a} showed,  for  matrices with
elements $X_{jk}$  which have a uniformly
sub exponential decay, i.e. condition \eqref{exptails} holds,
 the following result
\begin{align}
\Pr\Bigl\{\, \exists\,\, j:\,|\lambda_j-\gamma_j|\ge (\log n)^{C \operatorname{llog}_n }
\Big[\min\{(j,N-j+1)\Big]^{-\frac13}n^{-\frac23} \Bigr\} \quad\quad\quad\quad\notag\\ \le C\exp\{-(\log n)^{c \4 \operatorname{llog}_n}\}, \notag
\end{align}
for $n$ large enough.
It is straightforward to check that this bound implies that
\begin{equation}\label{rate}
\Pr\Bigl\{\sup_x|\mathcal F_n(x)-G(x)|\le Cn^{-1}(\log n)^{C \operatorname{llog}_n}\Bigr\}\ge 1- C\exp\{-(\log n)^{c \operatorname{llog}_n}\}.
\end{equation}
From the last inequality it is follows that $\E \Delta_n^*\le C\4 n^{-1}(\log n)^{C \4 \operatorname{llog}_n}$.
In this paper we derive   some improvement of the result \eqref{rate}
(reducing  the power of logarithm) using arguments
 similar to those used in \cite{ErdosYauYin:2010a}  and provide  a self-contained proof
 based on recursion  methods developed
in the papers of G\"otze and Tikhomirov \cite{GT:03}, \cite{GT:09} and \cite{T:09}.
It follows from the results of Gustavsson \cite{Gustavsson:2005} the best possible bound in the Gaussian case for
the rate of convergence in probability is $O(n^{-1}\sqrt{\log n})$.
For any positive constants $\alpha>0$  and $\varkappa>0$, define the quantities
\begin{equation}\label{beta}
 l_{n,\alpha} :=\log n\4(\log \log n)^{\alpha}
\quad \text{and} \quad
 \beta_n:=(l_{n,\alpha})^{\frac1{\varkappa}+\frac12}.
\end{equation}
The main result of this paper is the following
\begin{thm}\label{main} Let $\E X_{jk}=0$, $\E X_{jk}^2=1$.  Assume that
there exists a constant $\varkappa>0$ such that  inequality \eqref{exptails} holds, for any $1\le j\le k\le n$ and any $t\ge 1$.
Then, for  any positive $\alpha>0$ there exist  positive  constants $C$ and $c$ depending on $\varkappa $ and $\alpha$ only
such that
\begin{equation} \notag
\Pr\Big\{\sup_x|\mathcal F_n(x)-G(x)|>n^{-1}\beta_n^4\ln n\Big\}\le C\4 \exp\{-c \4 l_{n,\alpha} \}.
\end{equation}

\end{thm}
We apply the result of Theorem \ref{main} to study
the  eigenvectors of the matrix $\mathbf W$.
 Let $\mathbf u_j=(u_{j1},\ldots,u_{jn})^T$  be eigenvectors of the
matrix  $\mathbf W$ corresponding to the  eigenvalues $\lambda_j$, $j=1,\ldots,n$.
We prove the following result.
\begin{thm}\label{eigenvector} Under the conditions  of Theorem \ref{main}, for any positive $\alpha>0$,
  there exist positive  constants $C$ and $c$, depending on $\varkappa $ and $\alpha$ only
such that
\begin{equation}\label{deloc}
 \Pr\Bigl\{\max_{1\le j,k\le n}|u_{jk}|^2>\frac{\beta_n^2}n\Bigr\}\le C\exp\{- c\4 l_{n,\alpha}\},
\end{equation}
and
\begin{equation}\label{deloc1}
\Pr\Bigl\{\max_{1\le k\le n}\left|\sum_{\nu=1}^k|u_{j\nu}|^2-\frac kn\right|>\frac{\beta_n^2}{\sqrt n}\Bigr\}
\le C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}

\end{thm}

\section{Bounds for the  Kolmogorov distance between distribution functions  via  Stieltjes transforms}
To bound the error $\Delta_n^*$ we shall use an approach developed in previous work of the authors, see \cite{GT:03}.\\
We modify the bound of the  Kolmogorov distance between an arbitrary distribution function and the semi-circular distribution function
 via their Stieltjes transforms obtained in \cite{GT:03} Lemma 2.1. For $x\in[-2,2]$ define $\gamma(x)=:=2-|x|$.
 Given $\frac12>\varepsilon>0$ introduce the interval $\mathbb J_{\varepsilon}=\{x\in[-2,2]:\, \gamma(x)\ge\varepsilon\}$ and
$\mathbb J'_{\varepsilon}=\mathbb J_{\varepsilon/2}$.
For a distribution function $F$ denote by $S_F(z)$ its Stieltjes transform,
$$
S_F(z)=\int_{-\infty}^{\infty}\frac1{x-z}dF(x).
$$
\begin{prop}\label{smoothing}Let $v>0$ and $a>0$ and $\frac12>\varepsilon>0$ be positive numbers such that
\begin{equation} \label{constant}
 \frac1{\pi}\int_{|u|\le a}\frac1{u^2+1}du=\frac34=:\beta,
\end{equation}
and
\begin{equation}\label{avcond}
 2va\le \varepsilon^{\frac32}.
\end{equation}
If $G$ denotes the  distribution function of the standard semi-circular law, and $F$ is any distribution function,
 there exist some absolute constants $C_1$ and $C_2$ such that
\begin{align}
\Delta(F,G)&:= \sup_{x}|F(x)-G(x)|\notag\\&\le 2
\sup_{x\in\mathbb J'_{\varepsilon}}\Big|\im\int_{-\infty}^x(S_F(u+i\frac v{\sqrt{\gamma}})-S_G(u+i\frac v{\sqrt{\gamma}}))du\Big|+C_1v
+C_2\varepsilon^{\frac32}.  \notag
\end{align}

\end{prop}
\begin{rem}\label{rem2.2}
 For any $x\in\mathbb J_{\varepsilon}$ we have
$\gamma=\gamma(x)\ge\varepsilon$
and according to condition \eqref{avcond},
$\frac{av}{\sqrt\gamma}\le \frac{\varepsilon}2$.
\end{rem}

\begin{proof}
The proof of Proposition \ref{smoothing} is a straightforward adaptation  of  the proof of Lemma 2.1 from \cite{GT:03}. We include it here for the
sake of completeness. First we note that
\begin{align}
 \sup_x|F(x)-G(x)|&=\sup_{x\in[-2,2]}|F(x)-G(x)|=\max\Bigg\{\sup_{x\in\mathbb J_{\varepsilon}}|F(x)-G(x)|,\notag\\&
\sup_{x\in[-2,-2+\varepsilon]}|F(x)-G(x)|,\sup_{x\in[2-\varepsilon,2]}|F(x)-G(x)|\Bigg\}.
\end{align}
Furthermore, for $x\in[-2,-2+\varepsilon]$ we have
\begin{align}
 -G(-2+\varepsilon)\le F(x)-G(x)\le F(-2+\varepsilon)-G(-2+\varepsilon)+G(-2+\varepsilon)\notag\\
\le \sup_{x\in\mathbb J_{\varepsilon}}|F(x)-G(x)|+
G(-2+\varepsilon).
\end{align}
This inequality yields
\begin{equation}
\sup_{x\in[-2,-2+\varepsilon]}|F(x)-G(x)|\le \sup_{x\in\mathbb J_{\varepsilon}}|F(x)-G(x)|+G(-2+\varepsilon).
\end{equation}
Similarly we get
\begin{equation}
\sup_{x\in[2-\varepsilon,2]}|F(x)-G(x)|\le \sup_{x\in\mathbb J_{\varepsilon}}|F(x)-G(x)|+1-G(2-\varepsilon).
\end{equation}
Note that $G(-2+\varepsilon)=1-G(2-\varepsilon)$ and
$G(-2+\varepsilon)\le C\varepsilon^{\frac32}$  with some absolute constant $C>0$.
Combining all these relations  we get
\begin{equation}\label{gav1}
 \sup_x|F(x)-G(x)|\le\Delta_{\varepsilon}(F,G)+C\varepsilon^{\frac32},
\end{equation}
where $\Delta_{\varepsilon}(F,G)=\sup_{x\in\mathbb J_{\varepsilon}}|F(x)-G(x)|$.
We denote $v'=\frac v{\sqrt{\gamma}}$. For any $x\in \mathbb J'_{\varepsilon}$
\begin{align}\label{smoothm}
 \Big|\frac1{\pi}&\im\Big(\int_{-\infty}^x(S_F(u+iv')-S_G(u+iv'))du\Big)\Big|\notag\\&\ge \frac1{\pi}
\im\Big(\int_{-\infty}^x(S_F(u+iv')-S_G(u+iv'))du\Big)
\notag\\&=\frac1{\pi}\left[\int_{-\infty}^{x}\int_{-\infty}^{\infty}\frac{v'd(F(y)-G(y))}{(y-u)^2+{v'}^2}\right]du\notag\\&=
\frac1{\pi}\int_{-\infty}^x\left[\int_{-\infty}^{\infty}\frac{2v'(y-u)(F(y)-G(y))dy}{((y-u)^2+{v'}^2)^2}\right]\notag\\&=
\frac1{\pi}\int_{-\infty}^{\infty}(F(y)-G(y))\left[\int_{-\infty}^x\frac{2v'(y-u)}{((y-u)^2+{v'}^2)^2}du\right]dy\notag\\&=
\frac1{\pi}\int_{-\infty}^{\infty}\frac{F(x-v'y)-G(x-v'y)}{y^2+1} dy, \quad \text{by change of variables}.
\end{align}
Furthermore, using \eqref{constant} and the definition of $\Delta(F,G)$ we note that
\begin{equation}\label{int1}
\frac1{\pi} \int_{|y|>a}\frac{|F(x-v'y)-G(x-v'y)|}{y^2+1}dy\le (1-\beta)\Delta(F,G).
\end{equation}
Since $F$ is non decreasing, we have
\begin{align}
 \frac1{\pi}\int_{|y|\le a}&\frac{F(x-v'y)-G(x-v'y)}{y^2+1}dy\ge \frac1{\pi}\int_{|y|\le a}\frac{F(x-v'a)-G(x-v'y)}{y^2+1}dy
\notag\\&\ge
(F(x-v'a)-G(x-v'a))\beta\notag\\&\quad\q-\frac1{\pi}\int_{|y|\le a}|G(x-v'y)-G(x-v'a)|dy.
\end{align}
These inequalities together imply (using a change of variables in the last step)
\begin{align} \label{smoth1}
 \frac1{\pi}&\int_{-\infty}^{\infty}\frac{F(x-v'y)-G(x-v'y)}{y^2+1}dy\ge
\beta(F(x-v'a)-G(x-v'a))\notag\\&-\frac1{\pi}\int_{|y|\le a}|G(x-v'y)-G(x-v'a)|dy-(1-\beta)\Delta(F,G)\notag\\&\ge
\beta(F(x-v'a)-G(x-v'a))\notag\\&-\frac1{v'\pi}\int_{|y|\le v'a}|G(x-y)-G(x-v'a))|dy-(1-\beta)\Delta(F,G)
\end{align}
Note that according to Remark \ref{rem2.2}, $x\pm v'a\in\mathbb J'_{\varepsilon}$ for any $x\in\mathbb J_{\varepsilon}$.
 Assume first that $x_n\in\mathbb J_{\varepsilon}$ is a
sequence  such that
$F(x_n)-G(x_n)\to\Delta_{\varepsilon}(F,G)$. Then $x_n':=x_n+v'a\in\mathbb J'_{\varepsilon}$.
Using \eqref{smoothm} and \eqref{smoth1}, we get
\begin{align}\label{gav20}
 \sup_{x\in\mathbb J'_{\varepsilon}}&\left|\im\int_{-\infty}^x(S_F(u+iv')-S_G(u+iv'))du\right|\notag\\&\ge
\im\int_{-\infty}^{x_n'}(S_F(u+iv')-S_G(u+iv'))du\notag\\&\ge\beta(F(x_n'-v'a)-G(x_n'-v'a))\notag\\&\quad\q-\frac1{\pi v}
\sup_{x\in\mathbb J'_{\varepsilon}}{\sqrt{\gamma}}\int_{|y|\le 2v'a}|G(x+y)-G(x)|dy-(1-\beta)\Delta(F,G) \notag
\\&=\beta(F(x_n)-G(x_n))\notag\\&-\frac1{\pi v}
\sup_{x\in\mathbb J'_{\varepsilon}}{\sqrt{\gamma}}\int_{|y|<2v'a}|G(x+y)-G(x)|dy-(1-\beta)\Delta(F,G) .
\end{align}
Assume for definiteness  that $y>0$. Recal that $ \varepsilon\le 2\gamma$, for any $x\in\mathbb J_{\varepsilon}'$.
By Remark  \ref{rem2.2} with $\varepsilon/2$ instead $\varepsilon$, we have $0<y\le 2v'a\le\sqrt 2\varepsilon$, for any
$x\in\mathbb J'_{\varepsilon}$.
 For the semi-circular law we have,
\begin{align}
 |G(x+y)-G(x)|&\le y\sup_{u\in[x,x+y]}G'(u)\le y\4C\4\sqrt{\gamma+y}\notag\\&\le C\4 y\sqrt{\gamma+2v'a}
\le C\4 y\4\sqrt{\gamma+\varepsilon}\le Cy\sqrt{\gamma}.
\end{align}
This yields after integrating in $y$
\begin{align}\label{gav3}
\frac1{\pi v} \sup_{x\in\mathbb J'_{\varepsilon}}{\sqrt{\gamma}}\int_{0\le y\le 2v'a}|G(x+y)-G(x)|dy
\le\frac C{v} \sup_{x\in\mathbb J'_{\varepsilon}}\gamma{v'}^2\le Cv.
\end{align}
Similarly we get that
\begin{align}\label{gav3*}
\frac1{\pi v} \sup_{x\in\mathbb J'_{\varepsilon}}{\sqrt{\gamma}}\int_{0\ge y\ge -2v'a}|G(x+y)-G(x)|dy
\le\frac C{v} \sup_{x\in\mathbb J'_{\varepsilon}}\gamma{v'}^2\le Cv.
\end{align}
By inequality \eqref{gav1}
\begin{equation}\label{gt1}
 \Delta_{\varepsilon}(F,G)\ge \Delta(F,G)-C\varepsilon^{\frac32}.
\end{equation}
The  inequalities
 \eqref{gav20}, \eqref{gt1} and \eqref{gav3}, \eqref{gav3*} together yield as $n$ tends to infinity
\begin{align}
 \sup_{x\in\mathbb J'_{\varepsilon}}&\left|\im\int_{-\infty}^x(S_F(u+iv')-S_G(u+iv'))du\right|\notag\\&
\ge(2\beta-1)\Delta(F,G)-C\4v -C\varepsilon^{\frac32}
,\label{Deltaeps}
\end{align}
for some constant $C>0$. Similar arguments may be used to prove this inequality
in case  there is a sequence $x_n\in\mathbb J_{\varepsilon}$ such $F(x_n)-G(x_n)\to-\Delta_{\varepsilon}(F,G)$.
In view of \eqref{Deltaeps} and $2\beta-1=1/2$ this completes the proof.
\end{proof}
\begin{lem}\label{Cauchy}
 Under the conditions of Proposition \ref{smoothing}, for any $V>v$ and   $0<v\le \frac{\varepsilon^{3/2}}{2a}$
and $v'=v/\sqrt{\gamma}, \gamma= 2-|x|$, $x\in\mathbb J'_{\varepsilon}$
 as above, the following inequality holds
\begin{align}
 \sup_{x\in\mathbb J'_{\varepsilon}}&\left|\int_{-\infty}^x(\im(S_F(u+iv')-S_G(u+iv'))du\right|\notag\\&\le
\int_{-\infty}^{\infty}|S_F(u+iV)-S_G(u+iV)|du\notag\\&+
\sup_{x\in\mathbb J'_{\varepsilon}}\left|\int_{v'}^V\left(S_F(x+iu)-S_G(x+iu)\right)du\right|. \notag
\end{align}

\end{lem}
\begin{proof}Let $x\in \mathbb J'_{\varepsilon}$ be fixed. Let $\gamma=\gamma(x)$.
 Put $z=u+iv'$.   Since $v'=\frac v{\sqrt{\gamma}}\le \frac{\varepsilon}{2a}$, see \eqref{avcond},  we may assume without loss of
generality that $v'\le 4$
for  $x\in\mathbb J'_{\varepsilon}$.  Since the functions of $S_F(z)$ and $S_G(z)$ are analytic in the upper half-plane, it is enough to use Cauchy's theorem. We can write
 for $x\in\mathbb J'_{\varepsilon}$
\begin{equation} \notag
\int_{-\infty}^{x}\im(S_F(z)-S_G(z))du=\im\{\lim_{L\to\infty}\int_{-L}^x(S_F(u+iv')-S_G(u+iv'))du\}.
\end{equation}
By Cauchy's integral formula, we have
\begin{align}
 \int_{-L}^x(S_F(z)-S_G(z))du&=\int_{-L}^x(S_F(u+iV)-S_G(u+iV))du\notag\\&
+\int_{v'}^V(S_F(-L+iu)-S_G(-L+iu))du\notag\\&-\int_{v'}^V(S_F(x+iu)-S_G(x+iu))du. \notag
\end{align}
Denote by $\xi(\text{ resp. }\eta)$ a random variable with distribution function $F(x)$ (resp. $G(x)$). Then we have
\begin{equation} \notag
 |S_F(-L+iu)|=\left|\E\frac1{\xi+L-u}\right|\le {v'}^{-1}\Pr\{|\xi|>L/2\}+\frac2L,
\end{equation}
for any $v'\le u\le V$.
Similarly,
\begin{equation} \notag
 |S_G(-L+iu)|\le {v'}^{-1}\Pr\{|\eta|>L/2\}+\frac2L.
\end{equation}
These inequalities imply that
\begin{equation} \notag
\left|\int_{v'}^V(S_F(-L+iu)-S_G(-L+iu))du\right|\to 0\quad\text{as}\quad L\to\infty,
\end{equation}
which completes the proof.
\end{proof}
Combining the results of Proposition \ref{smoothing} and Lemma \ref{Cauchy}, we get
\begin{cor}\label{smoothing1}
 Under the conditions of Proposition \ref{smoothing} the following inequality holds
\begin{align}
 \Delta(F,G)&\le 2\int_{-\infty}^{\infty}|S_F(u+iV)-S_G(u+iV)|du+C_1v+C_2\varepsilon^{\frac32}\notag\\&
  + 2 \sup_{x\in\mathbb J'_{\varepsilon}}\int_{v'}^V|S_F(x+iu)-S_G(x+iu)|du,\notag
\end{align}
where $v'=\frac v{\sqrt{\gamma}}$ with $\gamma=2-|x|$ and $C_1,C_2 >0$ denote absolute constants.

\end{cor}

 We shall apply  the last inequality. We denote the Stieltjes transform of $\mathcal F_n(x)$
by $m_n(z)$ and the Stieltjes transform of the semi-circular law by $s(z)$. Let
$\mathbf R=\mathbf R(z)$ be the resolvent matrix of $\mathbf W$ given by
$\mathbf R=(\mathbf W-z\mathbf I_n)^{-1}$,
for all $z=u+iv$ with $v\ne 0$. Here and in what follows $\mathbf I_n$ denotes the identity matrix of dimension $n$.
Sometimes we shall omit the sub index in the notation of an identity  matrix.
It is well-known that the Stieltjes transform of the semi-circular distribution
satisfies the equation
\begin{equation}\label{stsemi}
s^2(z)+zs(z)+1=0
\end{equation}
(see, for example, equality (4.20) in \cite{GT:03}).
Furthermore, the
Stieltjes transform of an empirical  spectral distribution function $\mathcal F_n(x)$, say $m_n(z)$, is given by
\begin{equation}\notag m_n(z)=\frac1n\sum_{j=1}^n
R_{jj}=\frac1{2n}\Tr\mathbf  R.
\end{equation}
 (see, for instance, equality (4.3)
in \cite{GT:03}).
 Introduce the matrices $\mathbf W^{(j)}$,
which are  obtained from $\mathbf W$ by deleting the $j$-th row and the $j$-th column, and
the corresponding resolvent matrix $\mathbf R^{(j)}$ defined by
$\mathbf R^{(j)}:=(\mathbf W^{(j)}-zI_{n-1})^{-1}$ and let $m_n^{(j)}(z):=\frac1{n-1}\Tr\mathbf R^{(j)}.$
Consider the index sets $\mathbb T_j:=\{1,\ldots,n\}\setminus \{j\}$.
We shall use the representation
\be\notag
R_{jj}=\frac1{-z+\frac1{\sqrt
n}X_{jj}-\frac1n{{\sum_{k,l\in\mathbb T_j}}}X_{jk}X_{jl}R^{(j)}_{kl}},
\en
(see,
for example, equality (4.6) in \cite{GT:03}). We may rewrite it as
follows
\be\label{repr01}
 R_{jj}=-\frac1{z+m_n(z)}+
\frac1{z+m_n(z)}\varepsilon_jR_{jj}, \en
where
$\varepsilon_j:=\varepsilon_{j1}+\varepsilon_{j2}+\varepsilon_{j3}+\varepsilon_{j4}$ with
\begin{align}
&\varepsilon_{j1}:=\frac1{\sqrt n}X_{jj},\quad
\varepsilon_{j2}:=\frac1n{\sum_{k\in\mathbb T_j}}(X_{jk}^2-1)R^{(j)}_{kk},\notag\\
&\varepsilon_{j3}:=\frac1n{\sum_{k\ne
l\in\mathbb T_j}}X_{jk}X_{jl}R^{(j)}_{kl},\quad
\varepsilon_{j4}:=\frac1n(\Tr R^{(j)}-\Tr R).\label{epsjn}
\end{align}
This relation immediately implies the following two equations

\begin{align}
 R_{jj}=&-\frac1{z+m_n(z)}-\sum_{\nu=1}^3\frac{\varepsilon_{j\nu}}{(z+m_n(z))^2}+\notag\\& \quad\q\quad + \sum_{\nu=1}^3
\frac1{(z+m_n(z))^2}\varepsilon_{j\nu}\varepsilon_jR_{jj}+\frac1{z+m_n(z)}\varepsilon_{j4}R_{jj}, \notag
\end{align}

and
\begin{align}
 m_n(z)=&-\frac1{z+m_n(z)}-\frac1{(z+m_n(z))}\frac1n\sum_{j=1}^n\varepsilon_jR_{jj}\label{repR}\\
=&-\frac1{z+m_n(z)}-\frac1{(z+m_n(z))^2}\frac1n\sum_{\nu=1}^3\sum_{j=1}^n\varepsilon_{j\nu}+\notag\\&  +
\frac1{(z+m_n(z))^2}\frac1n\sum_{\nu=1}^3\sum_{j=1}^n\varepsilon_{j\nu}\varepsilon_{j}R_{jj}+
\frac1{z+m_n(z)}\frac1n\sum_{j=1}^n\varepsilon_{j4}R_{jj}. \label{repr03}
\end{align}

\section{Large deviations I}

In the following lemmas we shall bound $\varepsilon_{j\nu}$, for $\nu=1,\ldots,4$ and $j=1,\ldots,n$.
Using the exponential tails of the distribution of $X_{jk}$ we shall replace quantities like
e.g. $\E |X_{jk}|^p I(|X_{jk}| > \lna^{\frac1{\varkappa}})$ and others  by a uniform error bound $C\4 \exp\{-c\4 l_{n,\alpha}\}$
with  constants $C,c>0$ depending on $\varkappa$ and $\alpha$ varying from one instance to the next.
\begin{lem}\label{lem2}
 Assuming the conditions of  Theorem $\ref{main}$  there exist positive
 constants $C$ and $c$, depending on $\varkappa$ and $\alpha$ such that
\begin{equation}\notag
\Pr\{|\varepsilon_{j1}|\ge 2 \4l_{n,\alpha}^{\frac1{\varkappa}}\4n^{-\frac12}\}\le C\exp\{-c\4 l_{n,\alpha}\},
\end{equation}  for any $j=1,\ldots, n$.
\end{lem}
\begin{proof}
The result follows immediately from the hypothesis \eqref{exptails}.
\end{proof}

\begin{lem}\label{lem2.3}
Assuming the conditions of  Theorem \ref{main} we have, for any $z=u+iv$ with $v>0$ and any $j=1,\ldots,n$,
\begin{equation}\notag
|\varepsilon_{j4}|\le \frac1{nv}.
\end{equation}
\end{lem}
\begin{proof}
The conclusion of Lemma \ref{lem2.3} follows immediately from the obvious inequality $|\Tr\mathbf R-\Tr\mathbf R^{(j)}|\le v^{-1}$
(see Lemma 4.1 in \cite{GT:03}).
\end{proof}

\begin{lem}\label{largedev2}
 Assuming the  conditions of Theorem \ref{main}, for all $z=u+iv$ with $u\in\mathbb R$ and $v>0$, the following inequality
holds
\be\notag
\Pr\Bigl\{|\varepsilon_{j2}|> 3\4l_{n,\alpha}^{\frac2{\varkappa}+\frac12} n^{-\frac12}
(n^{-1}\sum_{l\in\mathbb T_j}|R^{(j)}_{ll}|^2)^{\frac12}\Bigr\}\le C\exp\{-c \4l_{n,\alpha}\},
\en
for some positive constants $c>0$ and $C$, depending on $\varkappa$ and $\alpha$ only.
\end{lem}
\begin{proof}
We use the following well-known inequality for sums of independent random variables.
Let $\xi_1,\ldots,\xi_n$ be independent random variables such that $\E\xi_j=0$ and $|\xi_j|\le \sigma_j$. Then,
for some numerical constant $c>0$,
\begin{equation}\label{largedev}
 \Pr\{|\sum_{j=1}^n\xi_j|>x\}\le c(1-\Phi(x/\sigma))\le\frac{c\sigma}x\exp\{-\frac{x^2}{2\sigma^2}\},
\end{equation}
where $\Phi(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^x\exp\{-\frac {y^2}2\}dy$ and $\sigma^2=\sigma_1^2+\cdots+\sigma_n^2$.
The last inequality holds for $x\ge\sigma$. (See, for instance \cite{Bentkus:2007},  p.1,  first inequality.)
We put
$\eta_l=X_{jl}^2-1$, and define,
$$
 \xi_l=\bigl(\eta_l\mathbb I\{|X_{jl}|\le  l_{n,\alpha}^{\frac1{\varkappa}} \}
-\E\eta_l\mathbb I\{|X_{jl}|\le  \lna^{\frac1{\varkappa}} \}\bigr) R^{(j)}_{ll}.
 $$

Note that $\E\xi_l=0$ and
$|\xi_l|\le 2l_{n,\alpha}^{\frac2{\varkappa}} |{R^{(j)}_{ll}|}.$
Introduce the $\sigma$-algebra $\textfrak M^{(j)}$ generated by the random variables $X_{kl}$ with $k,l\in\mathbb T_j$. Let $\E_j$ and ${\Pr}_j$
denote the conditional  expectation and the conditional probability with respect to $\textfrak M^{(j)}$. Note that the random variables $X_{jl}$
and the $\sigma$-algebra
$\textfrak M^{(j)}$ are independent.
Applying inequality \eqref{largedev} with
 $x:=l_{n,\alpha}^{\frac12}\sigma$ and with\newline
$\sigma^2=4nl_{n,\alpha}^{\frac4{\varkappa}} \Bigg(\frac1n\sum_{l\in\mathbb T_j}|{R^{(j)}_{ll}|^2}\Bigg)$,  we get
\begin{align} \label{previous}
\Pr\Bigl\{|\sum_{l\in \mathbb T_j}\xi_j|>x\Bigr\}&= \E{\Pr}_j\Bigr\{|\sum_{l\in \mathbb T_j}\xi_j|\ge x\Bigr\}\notag\\&\le
\E\exp\Bigl\{-\frac {x^2}{\sigma^2}\Bigr\}\le C\exp\{-c\4 l_{n,\alpha}\}.
\end{align}
Furthermore, note that
\begin{equation}\notag
 \E_j\eta_l\mathbb I\{|X_{jl}| \le  \lna^{\frac1{\varkappa}} \}=-\E_j\eta_l\mathbb I\{|X_{jl}| \ge  \lna^{\frac1{\varkappa}} \}.
\end{equation}
This implies
\begin{align}
 |\E_j\eta_l\mathbb I\{|X_{jl}| \le  \lna^{\frac1{\varkappa}} \}|&\le  \E_j^{\frac12}|\eta_l|^2
{\Pr}_j^{\frac12}\{|X_{jl}| > \lna^{\frac1{\varkappa}} \}\notag\\& \le  \E^{\frac12}|\eta_l|^2
\exp\{-\frac 12l_{n,\alpha} \}\le C\exp\{-\frac 12l_{n,\alpha} \}.\notag
\end{align}
The  last inequality implies that
\begin{align}\label{last}
 \Big|\frac1n\sum_{l\in\mathbb T_j}&\E_j\eta_l\mathbb I\{|X_{jl}|\le  \lna^{\frac1{\varkappa}})\}R^{(j)}_{ll}\Big|\notag\\&\le
\Bigg(\frac1n\sum_{l\in\mathbb T_j}| \E_j\eta_l\mathbb I\{|X_{jl}|\le  \lna^{\frac1{\varkappa}})\}|^2\Bigg)^{\frac12}
(\frac1n\sum_{l\in\mathbb T_j}|R^{(j)}_{ll}|^2)^{\frac12}\notag\\&\le
C\exp\{- cl_{n,\alpha}\}(\frac1n\sum_{l\in\mathbb T_j}|R^{(j)}_{ll}|^2)^{\frac12}.
\end{align}
Furthermore, we note that if $|X_{jl}| \le \lna^{\frac1{\varkappa}}$ for all $l\in\mathbb T_j$, (which holds with probability at
least $1-\varkappa^{-1}\exp\{ -c \4 l_{n,\alpha}\}$)
 \begin{align}\label{last1}
 |\varepsilon_{j2}|\le  \big|\frac1n\sum_{l\in\mathbb T_j}\xi_l|+
|\frac1n\sum_{l\in\mathbb T_j} \E_j\eta_l\mathbb I\{ |X_{jl}|\le \lna^{\frac1{\varkappa}} \} R^{(j)}_{ll}\big|.
 \end{align}

The inequalities \eqref{previous}, \eqref{last}  and \eqref{last1} together conclude the proof of Lemma \ref{largedev2}.
Thus  Lemma \ref{largedev2} is proved.

\end{proof}
\begin{cor}\label{cor01}
Assuming the conditions of Theorem \ref{main} for any $\alpha>0$  there exist positive  constants $c$ and $C$, depending on
$\varkappa$ and $\alpha$ such that for any $z=u+iv$ with $u\in\mathbb R$ and $v>0$
\begin{align}
 \Pr\{|\varepsilon_{j2}|>3 l_{n,\alpha}^{\frac2{\varkappa}+\frac12}
(nv)^{-\frac12}(\im\4  m_n^{(j)}(z))^{\frac12}\}\le C\exp\{-c\4 l_{n,\alpha}\}. \notag
\end{align}
\end{cor}
\begin{proof}Note that
\begin{equation} \notag
 n^{-1}\sum_{ l\in \mathbb T_j}|R^{(j)}_{ll}|^2\le n^{-1}\Tr|\mathbf R^{(j)}|^2=\frac1v\im m_n^{(j)}(z),
\end{equation}
where $|\mathbf R^{(j)}|^2=\mathbf R^{(j)}{\mathbf R^{(j)}}^*$.
The result follows now from Lemma \ref{largedev2}.

 \end{proof}

\begin{lem}\label{lem2.5}
 Assuming the conditions of Theorem $\ref{main}$, for any $j=1,\ldots,n$ and for any $z=u+iv$ with $u\in\mathbb R$ and $v>0$,
the following inequality holds,
\be \notag
\Pr\Bigl\{|\varepsilon_{j3}|> \beta_n^2
n^{-\frac12}(\frac1n\sum_{k\ne l\in\mathbb T_j}|R^{(j)}_{kl}|^2)^{\frac12} \Bigr\}\le C\exp\{-c \4 l_{n,\alpha}\}.
\en
\end{lem}
\begin{proof}
We shall use a large deviation bound for quadratic forms which follows from  results by Ledoux (see \cite{Ledoux:1996}).
\begin{prop}\label{quadraticform}
 Let $\xi_1,\ldots,\xi_n$ be independent random variables such that $|\xi_j|\le 1$. Let  $a_{ij}$ denote
 real numbers such that
$ a_{ij}= a_{ji}$ and $a_{jj}=0$. Let $Z=\sum_{l,k=1}^n \xi_l\xi_k a_{lk}$. Let $\sigma^2=\sum_{l,k=1}^n|a_{lk}|^2$.
Then for every $t>0$ there exists some positive constant $c>0$ such that the following inequality holds
\begin{align} \notag
 \Pr\{|Z|\ge\frac32\E^{\frac12}|Z|^2+t\}\le \exp\{-\frac{c\4 t}{\sigma}\}.
\end{align}
 \end{prop}
 \begin{Proof}
  Proposition \ref{quadraticform} follows from Theorem 3.1 in \cite{Ledoux:1996}.
 \end{Proof}
\begin{rem}
    Proposition \ref{quadraticform} holds for complex $a_{ij}$ as well. Here we should consider two quadratic forms with coefficients
$\re a_{jk}$ and $\im a_{jk}$.
   \end{rem}

In order to bound $\varepsilon_{j3}$ we use Proposition \ref{quadraticform} with
\begin{align}
\xi_l=\Big(X_{jl}\mathbb I\{|X_{jl}|\le \4 \lna^{\frac1{\varkappa}}\}
-\E X_{jl}\mathbb I\{|X_{jl}|\le  \lna^{\frac1{\varkappa}}\}
\Big)/(2\4 \lna^{\frac1{\varkappa}}).\notag
\end{align}
Note that the random variables $X_{jl}$, $l\in\mathbb T_j$ and the matrix $\mathbf R^{(j)}$ are mutually independent for any
fixed $j=1,\ldots,n$.
Moreover, we have
$|\xi_{l}|\le 1.$ Put $Z:=\sum_{k\ne l\in \mathbb T_j}\xi_l\xi_k R^{(j)}_{kl}$. Note that  $\mathbf R^{(j)}={\mathbf R^{(j)}}^T$. We have $\E_j|Z|^2=
2\sum_{k,l\in\mathbb T_j}|R^{(j)}_{kl}|^2$.
Applying Proposition \ref{quadraticform} with
$t= l_{n,\alpha}
(\sum_{l\ne k\in \mathbb T_j}|R^{(j)}_{lk}|^2)^{\frac12}$, we get
\begin{align}\label{covmat1.1}
 \E{\Pr}_j\Bigl\{|Z|\ge  l_{n,\alpha}
(\sum_{l\ne k\in \mathbb T_j}|R^{(j)}_{lk}|^2)^{\frac12}\Bigr\}\le C\exp\{-c\4 l_{n,\alpha}\}.
\end{align}
Furthermore,  for some appropriate $c>0$ and for $n\ge 2$
\begin{equation} \notag
 \Pr\{\exists j,l\in[1,\ldots,n]:\,|X_{jl}|>  \lna^{\frac1{\varkappa}}\}\le  \varkappa^{-1}
\4 n^2\exp\{-l_{n,\alpha}\} \le C\4 \exp\{-c\4l_{n,\alpha}\}
\end{equation}
and similarly since $\E X_{jl}=0$,
\begin{align}\label{covmat1.2}
& |\E X_{jl}\mathbb I\{|X_{jl}|\le  \lna^{\frac1{\varkappa}}\}|\notag\\
& \le
{\Pr}^{\frac12}\{\exists j,l\in[1,\ldots,n]:\,|X_{jl}|> \lna^{\frac1{\varkappa}}\}
\le C\exp\{-c\4 l_{n,\alpha}\}.
\end{align}
Introduce the random variables $\widehat \xi_l=X_{jl}\mathbb I\{|X_{jl}|\le \lna^{\frac1{\varkappa}}\}
/(2 \lna^{\frac1{\varkappa}})$ and $\widehat Z=\sum_{l,k\in\mathbb T_j}\widehat \xi_l\widehat \xi_kR^{(j)}_{lk}$.
Note that
\begin{align}\label{covmat1.3}
 \Pr\Bigl\{\sum_{l,k\in\mathbb T_j}X_{jk}X_{jl}R^{(j)}_{kl}\ne4l_{n,\alpha}^{\frac2{\varkappa}}\widehat Z\Bigr\}\le
C\exp\{-c\4 l_{n,\alpha}\}.
\end{align}
Furthermore, by \eqref{covmat1.2} we have
\begin{align}\label{covmat1.4}
|\frac  1n \sum_{l,k\in\mathbb T_j}R^{(j)}_{kl}
\E\widehat \xi_l\E\widehat \xi_k| \le C\exp\{-c\4 l_{n,\alpha}\}\4 \Big(\frac1n\sum_{k\ne l\in\mathbb T_j}|R^{(j)}_{kl}|^2\Big)^{\frac12}.
\end{align}
Finally, inequalities \eqref{covmat1.1}--\eqref{covmat1.4} together imply
\begin{align}
 \Pr\Bigl\{|\varepsilon_{j3}|> 4 \beta_n^2\4
n^{-\frac12}(\frac1n\sum_{k\ne l\in\mathbb T_j}|R^{(j)}_{kl}|^2)^{\frac12}\Bigr\}\le C\exp\{-c\4l_{n,\alpha}\}. \notag
\end{align}
Thus Lemma \ref{lem2.5} is proved.
\end{proof}

\begin{cor}\label{cor02}
 Under the conditions of Theorem \ref{main} there exist positive constants $c$ and $C$ depending on $\varkappa$ and
$\alpha$ such that for any $z=u+iv$ with $u\in\mathbb R$ and with $v>0$
\begin{align}
 \Pr\{|\varepsilon_{j3}|> 4 \beta_n^2\4
(nv)^{-\frac12}(\im m_n^{(j)}(z))^{\frac12}\}\le C\exp\{-c\4 l_{n,\alpha}\}. \notag
\end{align}
\end{cor}
\begin{proof}Note that as above
\begin{equation} \label{im1}
 n^{-1}\sum_{k\ne l\in \mathbb T_j}|R^{(j)}_{kl}|^2\le n^{-1} \Tr|\mathbf R^{(j)}|^2=\frac1v\im m_n^{(j)}(z).
\end{equation}
The result now follows from Lemma \ref{lem2.5}.

\end{proof}
To {summarize}  these results we recall
$\beta_n=(l_{n,\alpha})^{\frac1{\varkappa}+\frac12}$,
defined previously in \eqref{beta}. Without loss of generality we may assume that $\beta_n\ge 1$ and $l_{n,\alpha}\ge1$.
Then Lemmas \ref{lem2}, \ref{lem2.3}, Lemma \ref{largedev2} (with $l_{n,\alpha}^{\frac2{\varkappa}+\frac12}$ replaced by $\beta_n^2$),
and Lemma \ref{lem2.5} together imply
\begin{equation}\notag
 \Pr\Big\{|\varepsilon_{j}|>\frac{\beta_n^2}{\sqrt n}\Big(1+\frac{\im^{\frac12}m_n^{(j)}(z)}{\sqrt v}+\frac1{\sqrt v\sqrt{nv}}\Big)\Big\}\le
C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
 Using that
 \be \label{mnj}
 0<\im m_n^{(j)}(z)\le \im m_n(z)+\frac1{nv},
 \en we may rewrite the last inequality
\begin{equation}\label{important}
 \Pr\Big\{|\varepsilon_{j}|>\frac{\beta_n^2}{\sqrt n}\Big(1+\frac{\im^{\frac12}m_n(z)}{\sqrt v}+\frac1{\sqrt v\sqrt{nv}}\Big)\Big\}\le
C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
Denote by
\begin{equation}\label{definition}
 \Omega_n(z,\theta)=\Bigl\{\omega\in\Omega:\,|\varepsilon_{j}|\le
\frac{\theta\beta_n^2}{\sqrt n}\Big(1+\frac{\im^{\frac12}m_n(z)}{\sqrt v}+\frac{1}{\sqrt n v}\Big)\Bigr\},
\end{equation}
for any $\theta\ge1$.
Let
\begin{equation}\label{v0}
v_0:=\frac{d\beta_n^4}{n}
\end{equation}
 with a sufficiently large positive constant $d>0$.
We introduce the region $\mathcal D=\{z=u+iv\in\mathbb C: |u|\le 2,\, v_0<v\le 2\}$.  Furthermore,  we introduce the sequence $z_l=u_l+v_l$ in $\mathcal D$,
 recursively defined via
$u_{l+1}-u_l=\frac4{n^8}$ and $v_{l+1}-v_l=\frac2{n^8}$. Using a union bound, we have
\begin{equation} \label{zl}
 \Pr\{\cap_{z_l\in\mathcal D}\Omega_n(z_l,\theta)\}\ge1-C(\theta)\exp\{-c(\theta)\4 l_{n,\alpha}\}
\end{equation}
with some constant $C(\theta)$ and $c(\theta)$ depending on $\alpha, \varkappa$ and $\theta$.
Using the resolvent equality $\mathbf R(z)-\mathbf R(z')=-(z-z')\mathbf R(z)\mathbf R'(z)$, we get
\begin{equation}\notag
 |R^{(j)}_{k+n,l+n}(z)-R^{(j)}_{k+n,l+n}(z')|\le \frac{|z-z'|}{vv'}.
\end{equation}
This inequality and the definition of $\varepsilon_{j}$ together imply
 \begin{equation} \label{epsdiff}
 \Pr\Big\{|\varepsilon_j(z)-\varepsilon_j(z')|\le \frac{nl_{n,\alpha}^{\frac2{\varkappa}}|z-z'|}{v_0^2}\quad \text {for all } z,z'\in\mathcal D\Big \}\ge 1-C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
For any $z\in\mathcal D$ there exists a point $z_l$ such that $|z-z_l|\le Cn^{-8}$. This together with inequalities
\eqref{zl} and \eqref{epsdiff}  immediately implies that
\begin{align}\label{capprobab}
 \Pr\{\cap_{z\in\mathcal D}\Omega_n(z,2)\}&\ge\Pr\{\cap_{z_l\in\mathcal D}\Omega_n(z_l,1)\}-C\4\exp\{-c\4 l_{n,\alpha}\}\notag\\&\ge 1-C\exp\{-c\4 l_{n,\alpha}\},
\end{align}
 with some constants $C$ and $c$ depending on $\alpha$ and $\varkappa$ only. Let
\begin{equation}\label{Omega_n}
 \Omega_n:=\cap_{z\in\mathcal D}\Omega_n(z,2).
\end{equation}
Put now
\begin{equation}
  v_0':=v_0'(z)=\frac{\sqrt 2v_0}{\sqrt{\gamma}},
    \end{equation}
where $\gamma:=2-|u|$, $z=u+iv$ and $v_0$ is given by \eqref{v0}.
Note that $0\le \gamma\le2$, for $u\in[-2,2]$ and $v_0'\ge v_0$.
Denote
$\mathcal D':=\{z\in\mathcal D:\, v\ge v_0'\}$.
 \section{Bounds for $\abs{m_n(z)}$}
In this section we bound the probability that $\im m_n(z)\le C$ for some numerical constant $C$ and for any $z\in\mathcal D$.
We shall derive auxiliary bounds for the difference  between the Stieltjes transforms $m_n(z)$ of the empirical spectral measure
  of the matrix $\mathbf X$ and the Stieltjes transform $s(z)$ of the semi-circular law. Introduce the additional notations
\begin{align}
 \delta_n:=\frac1n\sum_{j=1}^n
\varepsilon_{j}R_{jj}. \notag
\end{align}
Recall that $s(z)$ satisfies the equation
\begin{equation}\label{semicircular0}
 s(z)=-\frac1{z+s(z)}.
\end{equation}
For the semi-circular law the following inequalities hold
\begin{equation}\label{sc1}
 |s(z)|\le 1\text{ and }|z+s(z)|\ge 1.
\end{equation}
Introduce  $g_n(z):=m_n(z)-s(z)$.
Equality \eqref{semicircular0}  implies that
\begin{equation}\label{semi20}
 1-\frac1{(z+s(z))(z+m_n(z))}=\frac{z+m_n(z)+s(z)}{z+m_n(z)}.
\end{equation}
The representation \eqref{repR}  implies
\begin{equation}\label{repr100}
 g_n(z)=\frac{g_n(z)}{(z+s(z))(z+m_n(z))}+\frac{\delta_n}{z+m_n(z)}.
\end{equation}
From here it follows by solving for $g_n(z)$ that
\begin{equation}\label{delta}
 g_n(z)=\frac{\delta_n(z)}{z+m_n(z)+s(z)}.
\end{equation}
\begin{lem}\label{simple}
 Let
\begin{equation}\label{imag}
 |g_n(z)|\le\frac 12.
\end{equation}
Then \quad
 $ |z+m_n(z)|\ge \frac12$
and
 $\im m_n(z) \le |m_n(z)|\le \frac32.$
\end{lem}
\begin{proof} This is an immediate consequence of inequalities \eqref{sc1} and of
\begin{equation}
 |z+m_n(z)| \ge|z+s(z)|-|g(z)|\ge \frac1 2,  \quad \text{ and } \quad
 |m_n(z)|  \le |s(z)|+|g(z)|\le \frac32.\notag
\end{equation}
\end{proof}
\vskip -0.7cm
\begin{lem}\label{lem1}
 Assume condition \eqref{imag} for $z=u+iv$ with $v\ge v_0$. Then for any $\omega\in  \Omega_n$, defined in \eqref{Omega_n}, we obtain
$ |R_{jj}|\le 4$.
\end{lem}
\vskip -0.7cm
\begin{proof}
By definition of $\Omega_n$  in \eqref{Omega_n}, we have
\begin{equation}
 |\varepsilon_j|\le \frac{\beta_n^2}{\sqrt n}\Big(1+\frac{\im^{\frac12}m_n(z)}{\sqrt v}+\frac1{\sqrt v\sqrt{nv}}\Big).
\end{equation}
Applying Lemmas \ref{simple} and \eqref{v0}, we get
 $|\varepsilon_j|\le \frac{A\beta_n^2}{\sqrt {nv}}$
 with some $A>0$ depending on the parameter $d\ge 1$ in $\eqref{v0}$ which we may choose such that
\begin{equation}\label{in01}
 |\varepsilon_j|\le \frac1{200},
\end{equation}
for any $\omega\in\Omega_n$, $n\ge2$, and $v\ge v_0$.
Using representation \eqref{repr01} and applying Lemma \ref{simple}, we get
 $|R_{jj}|\le 4$.
\end{proof}
\vskip -0.5cm
\begin{lem}\label{lem10}
 Assume condition \eqref{imag}. Then, for any $\omega\in\Omega_n$ and $v\ge v_0$,
\begin{equation}
 |g_n(z)|\le \frac1{100}.
\end{equation}
\end{lem}
\begin{proof}
 Lemma \ref{lem1}, inequality \eqref{in01}, and representation \eqref{delta} together imply
\begin{equation}
 |\delta_n|\le \frac{4}n\sum_{j=1}^n|\varepsilon_j|
\le \frac{4\beta_n^2}{\sqrt n}\Big(1+\frac{\im^{\frac12}m_n(z)}{\sqrt v}+\frac1{\sqrt v\sqrt{nv}}\Big)
\end{equation}
Note that
\begin{equation}\label{error20}
 |z+m_n(z)+s(z)|\ge\im z+\im m_n(z)+\im s(z)\ge \im(z+s(z))\ge\frac12 \im\{\sqrt{z^2-4}\}.
\end{equation}
For $z\in\mathcal D$ we get $\re(z^2-4)\le0$ and $\frac{\pi}2\le\arg(z^2-4)\le \frac{3\pi}2$.
Therefore,
\begin{equation}\label{error3}
 \im\{\sqrt{z^2-4}\}\ge\frac1{\sqrt 2}|z^2-4|^{\frac12}\ge \frac14\sqrt{\gamma+v},
\end{equation}
where $\gamma=2-|u|$.
These relations imply that
\begin{equation}
 \frac{|\delta_n|}{|z+m_n(z)+s(z)|}\le \frac{\beta_n^2}{\sqrt {nv}}+\frac{\beta_n^2}{\sqrt n\sqrt {v\sqrt{\gamma}}}
+\frac{\beta_n^2}{(nv)^{\frac32}\sqrt{\gamma}}.
\end{equation}
For $v\sqrt{\gamma}\ge v_0$, we get
\begin{equation}
 |g_n(z)|\le \frac{8\beta_n^2}{\sqrt {nv_0}}\le \frac1{100}
\end{equation}
by choosing the constant $d\ge1$ in $v_0$ appropriately large.
Thus the  lemma is proved.
\end{proof}
\begin{lem}\label{crucial0}
Assume that condition \eqref{imag} holds, for some $z=u+iv\in \mathcal D'$ and for any $\omega\in\Omega_n$,
(see \eqref{Omega_n} and the subsequent notions).
Then  \eqref{imag} holds as well for $z'=u+i\widehat v\in\mathcal D'$ with
  $v\ge \widehat v \ge v - n^{-8}$, for any $\omega\in\Omega_n$.
\end{lem}
\begin{proof}
First of all note that
\begin{equation} \notag
 |m_n(z)-m_n(z')|=\frac1n(v-\widehat v)|\Tr\mathbf R(z)\mathbf R(z')|\le \frac{v-\widehat v}{v\widehat v}\le \frac C{n^4}\le\frac1{100}
\end{equation}
and
$|s(z)-s(z')|\le \frac{|z-z'|}{v\widehat v}\le \frac1{100}$.
 By Lemma \ref{lem10}, we have
$|g_n(z)|\le \frac1{100}$.
All these inequalities together imply
$|g_n(z')|\le \frac3{100}<\frac12$.
Thus,  Lemma \ref{crucial0} is proved.
 \end{proof}
\begin{prop}\label{bound}
 Assuming the conditions of Theorem \ref{main} there exist constants $C>0$ and $c>0$ depending on $\varkappa$ and $\alpha$ only
such that
\begin{equation}
 \Pr\{|m_n(z)|\le \frac32 \text{ for any } z\in\mathcal D'\}\le C\exp\{-c\4l_{n,\alpha}\}.
\end{equation}
\end{prop}
\begin{proof}
 First we note that $|g_n(z)|\le \frac12\text{ a.s.}$, for $z=u+4i$.
By Lemma \ref{crucial0},
$|g_n(z')|\le\frac12 \text{ for any }\omega\in\Omega_n$.
Applying Lemma \ref{simple} and a union bound, we get
\begin{equation}
 \Pr\{|m_n(z)|\le \frac32 \text{ for any } z\in\mathcal D'\}\le C\exp\{-c\4l_{n,\alpha}\}.
\end{equation}
Thus the proposition is proved.
\end{proof}

\section{Large deviations II} In this Section we improve the bounds for $\delta_n$ We shall use   bounds for large deviation
probabilities of the sum of $\varepsilon_j$.
We start with
\begin{equation}\label{dn1}
\delta_{n1}=\frac1n\sum_{j=1}^n\varepsilon_{j1}.
\end{equation}
\begin{lem}\label{deltan1}
 There exist constants $c$ and $C$ depending on $\varkappa$ and $\alpha$ and such that
\begin{equation}\notag
\Pr\{|\delta_{n1}|>n^{-1}\beta_n\}\le C \exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
\end{lem}
\begin{proof}
 We repeat the proof of Lemma \ref{lem2}. Consider the truncated random variables
$\widehat X_{jj}=X_{jj}\mathbb I\{|X_{jj}|\le \4 \lna^{\frac1{\varkappa}}\}$.
By assumption \eqref{exptails},
\begin{equation}\notag
 \Pr\{|X_{jj}|> \4 \lna^{\frac1{\varkappa}}\}\le \varkappa^{-1}\exp\{-l_{n,\alpha}\}.
\end{equation}
Moreover,
\begin{equation}\notag
 |\E\widehat X_{jj}|\le C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
We define
$\widetilde X_{jj}=\widehat X_{jj}-\E\widehat X_{jj}$
and consider the sum
\begin{equation}\notag
\widetilde{\delta}_{n1}:=\frac1{n\sqrt n}\sum_{j=1}^n\widetilde X_{jj}.
\end{equation}
Since
$ |\widetilde X_{jj}|\le 2\4 \lna^{\frac1{\varkappa}}$,
by inequality \eqref{largedev}, we have
\begin{equation}\label{largedev11}
 \Pr\{|\widetilde{\delta}_{n1}|>n^{-1}l_{n,\alpha}^{\frac1{\varkappa}+\frac12}\}\le C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
Note that
\begin{equation} \notag
 |\widetilde{\delta}_{n1}-\delta_{n1}|\le \frac1n\sum_{j=1}^n|\E\widehat X_{jj}|\le C\4\exp\{-c \4 l_{n,\alpha}\}.
\end{equation}
This inequality and inequality \eqref{largedev11} together imply
\begin{equation}\notag
 \Pr\Bigl\{|\delta_{n1}|>n^{-1}l_{n,\alpha}^{\frac1{\varkappa}+\frac12}\Bigr\}\le C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
Thus,  Lemma \ref{deltan1} is proved.
\end{proof}
Consider now the quantity
\begin{equation} \label{dn2}
 \delta_{n2}:=\frac1{n^2}\sum_{j=1}^n\sum_{l\in\mathbb T_j}(X_{jl}^2-1)R^{(j)}_{ll}.
\end{equation}
We prove the following lemma
\begin{lem}\label{deltan2}Let $v_0=\frac{d\beta_n^4}n$ with some numerical constant $d\ge1$.
 Under the conditions of Theorem \ref{main} there exist  constants $c$ and $C$, depending on $\varkappa$ and $\alpha$ only,   such that
\begin{equation} \notag
 \Pr\{|\delta_{n2}|> 2n^{-1} \beta_n^2\frac1{\sqrt v}({\frac32}+\frac1{nv})^{\frac12}\}\le C\exp\{-c\4 l_{n,\alpha}\},
\end{equation}
 for any $z\in\mathcal D'$.
\end{lem}
\begin{proof}
Introduce the truncated random variables
$\xi_{jl}={\widehat X}_{jl}^2-\E{\widehat X}_{jl}^2$,
where $\widehat X_{jl}=X_{jl}\mathbb I\{|X_{jl}|\le \4 \lna^{\frac1{\varkappa}}\}$.
It is straightforward to check that
\begin{equation}\label{centr}
 0\le 1-E{\widehat X}_{jl}^2\le C\exp\{-c \4 l_{n,\alpha}\}.
\end{equation}
We shall need the following quantities as well
\begin{align}
 \widehat{\delta}_{n2}=\frac1{n^2}\sum_{j=1}^n\sum_{l\in\mathbb T_j}(\widehat X_{jl}^2-1)R^{(j)}_{ll} \quad \text{and}\quad
\widetilde{\delta}_{n2}=\frac1{n^2}\sum_{j=1}^n\sum_{l\in\mathbb T_j}\xi_{jl}R^{(j)}_{ll}. \notag
\end{align}
By assumption \eqref{exptails},
\begin{align}
\Pr\{\delta_{n2}\ne\widehat{\delta}_{n2}\}\le \sum_{j=1}^n\sum_{l\in\mathbb T_j}\Pr\{|X_{jl}|>\4 \lna^{\frac1{\varkappa}}\}\le C\exp\{-c\4 l_{n,\alpha}\} .\notag
\end{align}
By inequality \eqref{centr},
\begin{align}
 |\widehat{\delta}_{n2}-\widetilde{\delta}_{n2}|&\le \frac1{n^2}\sum_{j=1}^n\sum_{l\in\mathbb T_j}|\E{\widehat X}_{jl}^2-1||R^{(j)}_{ll}|\le
 Cv_0^{-1}\exp\{-c \4 l_{n,\alpha}\}\notag\\&\le C\exp\{-c\4l_{n,\alpha}\},\notag
\end{align}
 for $v\ge v_0$ and $C,c>0$ which are independent of $d\ge1$. \newline
Let
$\zeta_j:=\frac1{\sqrt n}\sum_{l\in\mathbb T_j}\xi_{jl}R^{(j)}_{ll}$.
Then
$\widetilde{\delta}_{n2}=\frac1{n^{\frac32}}\sum_{j=1}^n\zeta_j$.
Let $\textfrak N_j$, for $j=1,\ldots n$, denote the $\sigma$-algebras generated by the random variables $X_{lk}$ with $1\le l\le j$ and $1\le k\le j$.
Let $\textfrak N_0$ denote  the trivial $\sigma$-algebra.
Note that the sequence $\widetilde{\delta}_{n2}$ is a  martingale with respect to the  $\sigma$--algebras $\textfrak N_j$. In fact,
\begin{equation} \notag
 \E\{\zeta_j|\textfrak N_{j-1}\}=\E\{\E\{\zeta_j|\textfrak M^{(j)}\}|\textfrak N_{j-1}\}=0.
\end{equation}
In order  to use large deviation bounds for $\widetilde{\delta}_{n2}$ we replace the differences $\zeta_j$ by truncated random variables.
We put
$\widehat\zeta_j=\zeta_j\mathbb I\{|\zeta_j|\le l_{n,\alpha}^{\frac2{\varkappa}+\frac12}
(\frac32+\frac1{nv})^{\frac12}\}$. Denote by $t_{nv}^2=\frac32+\frac1{nv}$.
Since $\zeta_j$ is a sum of independent bounded random variables with mean zero (conditioned on $\textfrak M^{(j)}$), similar as in  Lemma
\eqref{largedev2} we get
\begin{equation} \notag
 {\Pr}_j\Bigl\{|\zeta_j|> l_{n,\alpha}^{\frac2{\varkappa}+\frac12}
(\frac1n\sum_{l\in\mathbb T_j}|R^{(j)}_{ll}|^2)^{\frac12}\Bigr\}\le C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
Using \eqref{im1} and \eqref{mnj}, we have
\begin{equation}\label{140}
 \frac1n\sum_{l\in\mathbb T_j}|R^{(j)}_{ll}|^2\le \frac1vt_{nv}^2.
\end{equation}
By Proposition \ref{bound}, we have
\begin{equation}
 {\Pr}_j\Bigl\{|\zeta_j|> l_{n,\alpha}^{\frac2{\varkappa}+\frac12}v^{-\frac12}
t_{nv}\Bigr\}\le C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
This implies that
\begin{equation}\label{lem2ineq1}
 \Pr\{\sum_{j=1}^n\zeta_j\ne \sum_{j=1}^n\widehat\zeta_j\}\le C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
Furthermore, introduce the random variables
$\widetilde{\zeta}_{j}=\widehat\zeta_j-\E\{\widehat\zeta_j|\textfrak N_{j-1}\}$.
 First we note that
\begin{equation}\notag
 \E\{\widehat\zeta_j|\textfrak N_{j-1}\}=-\E\Big\{\zeta_j\mathbb I\{|\zeta_j|> l_{n,\alpha}^{\frac2{\varkappa}+\frac12}v^{-\frac12}
t_{nv}\}\Big|\textfrak N_{j-1}\Big\}.
\end{equation}
Applying Cauchy-Schwartz,  $E_j\xi_{jl}\xi_{jl'} R^{(j)}_{ll}R^{(j)}_{l'l'} =0$ for $l\ne l',\,l,l'\in \mathbb T_j$ and
 $|R^{(j)}_{ll}|\le v^{-1}$  as well as   $\E\{\E_j\{|\zeta_j|^2\}|\textfrak N_{j-1}\}\le\frac1{nv}\sum_{l\in\mathbb T_j}\E |\xi_{jl}|^2 $
 we get
\begin{align}\label{lem2ineq2}
 |\E&\{\widehat\zeta_j|\textfrak N_{j-1}\}|\le C\E^{\frac12}\{|\zeta_j|^2|\textfrak N_{j-1}\}
{\Pr}^{\frac12}\Big\{|\zeta_j|> l_{n,\alpha}^{\frac2{\varkappa}+\frac12}v^{-\frac12}
t_{nv}\}\Big|\textfrak N_{j-1}\Big\}\notag\\&
= C\4 \E^{\frac12}\{\E_j\{|\zeta_j|^2\}|\textfrak N_{j-1}\}
\E^{\frac12}\Big\{{\Pr}_j\Big\{|\zeta_j|> l_{n,\alpha}^{\frac2{\varkappa}+\frac12}v^{-\frac12}
t_{nv}\}\Big\}\Big|\textfrak N_{j-1}\Big\}
\notag\\&\le
Cv^{-1}\Big(\frac1n\sum_{l\in\mathbb T_j}\E|\xi_{jl}|^2\Big)^{\frac12}\exp\{-c\4 l_{n,\alpha}\}
\le C\exp\{-c\4 l_{n,\alpha}\},
\end{align}
for $v\sqrt{\gamma}\ge v_0$ with constants $C$ and $c$ depending on $\alpha$ and $\varkappa$.

Furthermore, we may use a martingale bound due to Bentkus, \cite{Bentkus:2007}, Theorem 1.1. It provides the following result.
Let $\textfrak N_0=\{\emptyset,\Omega\}\subset\textfrak N_1\subset\cdots\subset\textfrak N_n\subset\textfrak M$ be a family of
 $\sigma$-algebras of a measurable space $\{\Omega,\textfrak M\}$. Let $M_n=\xi_1+\ldots +\xi_n$ be a martingale with bounded differences
$\xi_j=M_j-M_{j-1}$ such that
$ \Pr\{|\xi_j|\le b_j\}=1, \quad\text{for}\quad j=1,\ldots,n$.
Then, for $x>\sqrt8$
\begin{equation} \notag
 \Pr\{|M_n|\ge x\}\le c\4(1-\Phi(\frac x{\sigma}))=\int_{\frac x{\sigma}}^{\infty}\varphi(t)dt, \quad\varphi(t)=\frac1{\sqrt{2\pi}}
\exp\{-\frac{t^2}2\},
\end{equation}
with some numerical constant $c>0$ and $\sigma^2=b_1^2+\cdots+b_n^2.$
Note that for  $t>C$
$$
1-\Phi(t)\le \frac1C\varphi(t).
$$
Thus, this leads to the inequality
\begin{equation}\label{bentkus}
 \Pr\{|M_n|\ge x\}\le \exp\{-\frac{x^2}{2\sigma^2}\},
\end{equation}
which we shall use  to bound $\widetilde{\delta}_{n2}$.
Take $M_n=\sum_{j=1}^n\widetilde{\delta_j}$ with $|\widetilde{\delta_j}|$ bounded by
$b_j=2l_{n,\alpha}^{\frac2{\varkappa}+\frac12}v^{-\frac12}t_{nv}$.
By Proposition \ref{bound} obtain
\begin{equation}\label{13}
 \sigma^2=4nv^{-1}l_{n,\alpha}^{\frac4{\varkappa}+1}t_{nv}^2.
\end{equation}
Inequalities \eqref{bentkus} with $x=l_{n,\alpha}^{\frac12}\sigma$ and \eqref{13} together imply
\begin{equation}\label{lem2ineq3}
 \Pr\{|\widetilde{\delta}_{n2}|> 2n^{-1}\beta_n^2\frac1{\sqrt v}t_{nv}\}
\le  C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
Inequalities \eqref{lem2ineq1}--\eqref{lem2ineq3} together conclude the proof of Lemma \ref{deltan2}.
\end{proof}
Let
\begin{equation} \label{dn3}
 \delta_{n3}:=\frac1{n^2}\sum_{j=1}^n\sum_{l\ne k\in \mathbb T_j}X_{jl}X_{jk}R^{(j)}_{lk}.
\end{equation}
\begin{lem}\label{deltan3}Let $v_0=\frac{d\beta_n^4}n$ with some numerical constant $d>1$. Under condition of Theorem \ref{main}
 there exist constants $c$ and $C$, depending on $\varkappa$,  $\alpha$ only such that
\begin{equation} \notag
 \Pr\{|\delta_{n3}|> \frac{4\beta_n^2l_{n,\alpha}^{\frac12}}{n \sqrt v}(\frac32+\frac1{nv})^{\frac12}\}\le C\exp\{-c\4 l_{n,\alpha}\},
\end{equation}
for any $z\in\mathcal D'$.
\end{lem}
\begin{proof}
The proof of this Lemma is similar to the proof of Lemma \ref{deltan2}. We introduce the random variables
$\eta_j=\frac1{ n}\sum_{l\ne k\in \mathbb T_j}X_{jk}X_{jl}R^{(j)}_{lk}$
and note that the sequence
   $M_j=\frac1n\sum_{m=1}^j\eta_m$
is martingale with respect to the $\sigma$--algebras $\textfrak N_j$, for $j=1,\ldots, n$.
By Proposition \ref{bound}, using inequality \eqref{140}, we get
\begin{equation}
 \Pr\{\frac1n\sum_{l,k\in\mathbb T_j}|R^{(j)}_{lk}|^2\le \frac1vt_{nv}^2\text{ for any } z\in\mathcal D'\}\ge 1-C\exp\{-c\4l_{n,\alpha}\}.
\end{equation}
At first  we apply Proposition \ref{quadraticform}  replacing
$\eta_j$ by truncated random variables and then
apply  the martingale bound of Bentkus \eqref{bentkus}.
Introduce the random variables $\widehat X_{jk}=X_{jk}\mathbb I\{|X_{jk}|\le l_{n,\alpha}^{\frac1{\varkappa}}\}$ and
$\widetilde X_{jk}=\widehat X_{jk}-\E\widehat X_{jk}$. By condition \eqref{exptails}, we have
\begin{equation}\label{one}
 \Pr\{X_{jk}\ne \widehat X_{jk}\}\le C\exp\{-cl_{n,\alpha}\}.
\end{equation}
The same condition jelds
\begin{equation}\label{two}
 |\E\widehat X_{jk}|=|\E X_{jk}\mathbb I\{|X_{jk}|> l_{n,\alpha}^{\frac1{\varkappa}}\}|\le C\exp\{-cl_{n,\alpha}\}
\end{equation}
Let
\begin{equation}
 \widehat \eta_j=\frac1n\sum_{l\ne k\in \mathbb T_j}\widehat X_{jk}\widehat X_{jl}R^{(j)}_{lk},\text{ and }
\widetilde \eta_j=\frac1n\sum_{l\ne k\in \mathbb T_j}\widetilde X_{jk}\widetilde X_{jl}R^{(j)}_{lk}.
\end{equation}
Inequality \eqref{one} implies that
\begin{equation}
 \Pr\{\eta_j\ne\widehat\eta_j\}\le C\exp\{-cl_{n,\alpha}\}.
\end{equation}
Inequality \eqref{two} and $|\widetilde X_{jk}|\le 2l_{n,\alpha}$ together imply
\begin{equation}
 \Pr\{|\widehat \eta_j-\widetilde \eta_j|\le Cl_{n,\alpha}^{\frac1{\varkappa}}\exp\{-c\4 l_{n,\alpha}\}v^{-\frac12}t_{nv}\} =1.
\end{equation}
Applying now Propositions \ref{bound} and \ref{quadraticform}, and inequality \eqref{140}, similar to Lemma \ref{lem2.5} we get,
introducing  $r_{v,n}:= v^{-\frac12}\beta_n^2t_{nv}$,
\begin{equation}
 \Pr\{|\widetilde\eta_j|>n^{-\frac12} r_{v,n}
 \}\le C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
Now we introduce
\begin{align}
 \theta_j=\eta_j\mathbb  I\{|\eta_j|\le n^{-\frac12}  r_{v,n}
 \}  -\E\eta_j\mathbb I\{|\eta_j|\le n^{-\frac12}r_{v,n}\}.
\end{align}
Furthermore, we consider the random variables
$\widetilde \theta_j=\theta_j-\E\{\theta_j|\textfrak N_{j-1}\}$. The sequence $\widehat M_s$, defined by
 $\widehat M_s=\sum_{m=1}^s\widetilde\theta_m$,
is a martingale with respect to the $\sigma$-algebras $\textfrak N_s$, for $s=1,\ldots, n$.
Similar to the proof of Lemma \ref{deltan1} we get
\begin{align}
 \Pr\{|\widehat M_n-M_n|> 4l_{n,\alpha}^{\frac12}r_{v,n}
 \}\le C\exp\{-cl_{n,\alpha}\}.
\end{align}
Applying inequality \eqref{bentkus} to $\widehat M_n$ with $\sigma^2=16 r_{v,n}^2
$ and $x=l_{n,\alpha}^{\frac12}\sigma$, we get
\begin{align}
 \Pr\{|\widehat M_n|> 4l_{n,\alpha}^{\frac12}r_{v,n}
 \}\le C\exp\{-cl_{n,\alpha}\}.
\end{align}
Thus the  Lemma is proved.
\end{proof}
Finally, we shall bound
\begin{equation} \label{dn4}
\delta_{n4}:=\frac1{n^2}\sum_{j=1}^n(\Tr\mathbf R-\Tr\mathbf R^{(j)})R_{jj}.
\end{equation}
\begin{lem}\label{deltan4}
 For any $z=u+iv$ with $v>0$ the following inequality
\begin{equation}
|\delta_{n4}|\le \frac1{nv}\im m_n(z) \text{ a. s.}
\end{equation}
holds.
\end{lem}
\begin{proof}
 By formula (5.4) in \cite{GT:03}, we have
\begin{align}
 (\Tr\mathbf R-\Tr\mathbf R^{(j)})R_{jj}=(1+\frac1n\sum_{l,k\in T_j}X_{jl}X_{jk}(R^{(j)})^2_{lk})R_{jj}^2
=\frac{d}{dz}R_{jj}.
\end{align}
From here it follows that
\begin{equation}
 \frac1{n^2}\sum_{j=1}^n(\Tr\mathbf R-\Tr\mathbf R^{(j)})R_{jj}=\frac1{n^2}\frac{d}{dz}\Tr\mathbf R=\frac1{n^2}\Tr \mathbf R^2.
\end{equation}
Finally, we note that
\begin{equation} \notag
 |\frac1{n^2}\Tr \mathbf R^2|\le \frac1{nv}\im m_n(z).
\end{equation}
The last inequality concludes the proof.
Thus,  Lemma \ref{deltan4} is proved.
\end{proof}

\section{Stieltjes transforms} We shall derive auxiliary bounds for the difference  between the Stieltjes transforms $m_n(z)$ of the empirical
spectral measure
  of the matrix $\mathbf X$ and the Stieltjes transform $s(z)$ of the semi-circular law. Recalling the definitions of
 $\varepsilon_j, \varepsilon_{j\mu}$ in \eqref{epsjn} and of $\delta_{n\nu}$   in \eqref{dn1},\eqref{dn2}, \eqref{dn3} as well as \eqref{dn4},
 we introduce the additional notations
\begin{align}
 \delta_n':=\delta_{n1}+\delta_{n2}+\delta_{n3},\quad\widehat{\delta}_n:=\delta_{n4},\quad \overline {\delta}_n:=\frac1n\sum_{\nu=1}^3
\sum_{j=1}^n
\varepsilon_{j\nu}\varepsilon_jR_{jj}. \label{dnp}
\end{align}
Recall that $g_n(z):=m_n(z)-s(z)$.
The representation \eqref{repr03}  implies
\begin{equation}\label{repr10}
 g_n(z)=\frac{g_n(z)}{(z+s(z))(z+m_n(z))}-\frac{\delta_n'}{(z+m_n(z))^2}+\frac{\widehat{\delta}_n}{z+m_n(z)}+\frac{\overline{\delta}_n}
{(z+m_n(z))^2}.
\end{equation}
The equalities  \eqref{repr10} and  \eqref{semi20} together yield
\begin{align}\label{repr101}
 |g_n(z)|&\le \frac{|\delta_n'|+|\overline{\delta}_n|}{|z+m_n(z)||z+s(z)+m_n(z)|}+\frac{|\widehat{\delta}_n|}{|z+s(z)+m_n(z)|}.
\end{align}
For any $z\in\mathcal D'$ introduce the events
\begin{align}
 \widehat{\Omega}_n(z)&:=\{\omega\in\Omega:\,|{\delta}'_n|\le
\Bigl(\frac{\beta_n}n+\frac{\beta_n^2l_{n,\alpha}^{\frac12}\sqrt{\frac32}}{n\sqrt v}+\frac{\beta_n^2l_{n,\alpha}^{\frac12}}{n^{\frac32}v}\Bigr)\}
,\\
\widetilde {\Omega}_n(z)&:=
\{\omega\in\Omega:\,|\widehat{\delta_n}|\le\frac {C\4 \im m_n(z)}{nv}\},\\
\overline{\Omega}_n(z)&:=\{\omega\in\Omega:\,|\overline\delta_n|\le(\frac{\beta_n^2}{n}+\frac{\beta_n^4(\im m_n(z)+\frac1{nv})}{nv}+\frac1{n^2v^2})
\frac1n\sum_{j=1}^n|R_{jj}|\}. \notag
\end{align}
Put $\Omega_n^*(z):=\widehat{\Omega}_n(z)\cap\widetilde {\Omega}_n(z)\cap\overline{\Omega}_n(z)$.
By Lemmas \ref{deltan1}--\ref{deltan3}, we have
\begin{equation} \notag
 \Pr\{\widehat{\Omega}_n(z)\}\ge1-C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
The proof of the last relation is similar to the proof of inequality \eqref{capprobab}.
By Lemma \ref{deltan4},
\begin{equation} \notag
 \Pr\{\widetilde {\Omega}_n(z)\}=1.
\end{equation}
Note that
\begin{equation} \notag
 |\varepsilon_{j\nu}\varepsilon_{j4}|\le \frac12(|\varepsilon_{j\nu}|^2+|\varepsilon_{j4}|^2).
\end{equation}
By Lemmas \ref{largedev2} and \ref{lem2.5}, we have, for $\nu=2,3$,
\begin{equation} \notag
 \Pr\Bigl\{|\varepsilon_{j\nu}|^2>\frac{\beta_n^{4}}{ nv}({\im m_n(z)}+\frac{1}{n v})\Bigr\}\le C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
According to Lemma \ref{lem2},
\begin{equation}
 \Pr\Bigl\{|\varepsilon_{j1}|^2>\frac{\beta_n^2}{ n}\}\le C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
and, by Lemma  \ref{lem2.3}
\begin{equation}\notag
\Pr\{|\varepsilon_{j4}|^2\le \frac1{n^2v^2}\}=1.
\end{equation}
Similarly as in \eqref{capprobab} we may show that
\begin{equation} \notag
 \Pr\{\cap_{z\in\mathcal D}\Omega_n^*(z)\cap\Omega_n\}\ge1-C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
Let
\begin{equation} \notag
 \Omega_n^*:=\cap_{z\in\mathcal D}\Omega_n^*(z)\cap\Omega_n,
\end{equation}
 where $\Omega_n$ was defined in \eqref{Omega_n}.
 We   prove now some auxiliary  lemmas.

\begin{lem}\label{lem2.6} Let $z=u+iv\in\mathcal D$ and $\omega\in\Omega_n^*$. Assume that
\begin{equation}\label{cond1}
 |g_n(z)|\le \frac12.
\end{equation}
Then
the following bound
holds
\begin{equation}\notag
|g_n(z)|\le \frac{C\beta_n^{4}}{nv}+\frac{C\beta_n^4}{n^2v^2\sqrt{\gamma+v}}.
\end{equation}
\end{lem}
\begin{proof}
Inequality \eqref{repr101} implies that for $\omega\in\Omega_n^*$
\begin{align}\label{bound1}
 |g_n(z)|&\le\frac{\beta_n^{2}l_{n,\alpha}^{\frac12}(1+\sqrt{\frac32})}{n\sqrt{v}|z+m_n(z)||z+s(z)+m_n(z)|}+\frac{ C\4  \im m_n(z)}
{n v|z+s(z)+m_n(z)|}\notag\\&\quad\q + \frac{ \beta_n^2l_{n,\alpha}^{\frac12}}
{n^{\frac32} v|z+m_n(z)||z+s(z)+m_n(z)|}\notag\\&\quad\q+
\frac{\beta_n^{4}}{nv|z+m_n(z)||z+s(z)+m_n(z)|}\Big(\im m_n(z)+\frac{1}{nv}\Big)\frac1n\sum_{j=1}^n|R_{jj}|.
\end{align}
Inequality  \eqref{bound1} and Lemmas \ref{simple}, inequalities  \eqref{error20}, \eqref{error3} together imply
\begin{align}\label{bound1a}
 |g_n(z)|\le\frac {C\beta_n^{4}}{nv}\Big(1+\frac{1}{nv\sqrt{\gamma+v}}\Bigr).
\end{align}
This inequality completes the proof of lemma.
\end{proof}
Put now
\begin{equation}
  v_0':=v_0'(z)=\frac{\sqrt 2v_0}{\sqrt{\gamma}},
    \end{equation}
where $\gamma:=2-|u|$, $z=u+iv$ and $v_0$ given by \eqref{v0}.
Note that $0\le \gamma\le2$, for $u\in[-2,2]$ and $v_0'\ge v_0$.
Denote
$\widehat {\mathcal D}:=\{z\in\mathcal D:\, v\ge v_0'\}$.
\begin{cor}\label{cor3}
 Assume that $ |g_n(z)|\le \frac12$, for  $\omega\in\Omega_n^*$ and  $z=u+iv\in\widehat {\mathcal D}$.
 \\
Then
$
|g_n(z)|\le \frac1{100},
$
for sufficiently large $d$ in the definition of $v_0$.
\end{cor}
\begin{proof} Note that for $v\ge v_0'$
\begin{equation} \label{defd}
\frac {C\beta_n^{4}}{nv}+\frac{C\beta_n^4}{n^2v^2\sqrt{\gamma+v}}\le \frac{C\sqrt{\gamma}}{d}+
\frac{C\sqrt{\gamma}}{d^2\beta_n^4}\le\frac1{100},
\end{equation}
for an appropriately  large  constant $d$ in the definition of $v_0$.
 Thus, the Corollary is proved.
\end{proof}
\begin{rem}
 In what follows we shall assume that $d\ge 1$ is chosen and fixed such that inequality \eqref{defd} holds.
\end{rem}

Assume that $N_0$ is sufficiently large number such that for any $n\ge N_0$ and for any $v\in\mathcal D$ the right hand side of
inequality  \eqref{bound1a} is smaller then $\frac1{100}$.
In the what follows we shall assume that $n\ge N_0$ is fixed.
We repeat here Lemma \ref{crucial0}.
It is similar to Lemma 3.4 in \cite{ErdosYauYin:2010}.
\begin{lem}\label{crucial}
Assume that condition \eqref{cond1} holds, for some $z=u+iv\in \mathcal D'$ and for any $\omega\in\Omega_n^*$.
Then   eqref{cond1}  holds for $z'=u+i\widehat v\in\mathcal D$  as well with
  $v\ge \widehat v \ge v - n^{-8}$, for any $\omega\in\Omega_n^*$.
\end{lem}
\begin{proof}To prove this Lemma is enough to repeat the proof of Lemma \ref{crucial0}.
\end{proof}
 \vskip -0.7cm
 \begin{prop}\label{prop10}
 There exist positive constants $C$, $c$, depending on $\alpha$ and $\varkappa$ only such that
\begin{align}\label{mainbound}
 \Pr\Big\{|g_n(z)|>\frac {C\beta_n^{4}}{nv}+\frac{C\beta_n^4}{n^2v^2\sqrt{\gamma+v}} \Big\}\le
C\exp\{-c\4l_{n,\alpha}\}.
\end{align}
 for all $z\in\mathcal D'$
\end{prop}
\begin{proof}
 Note that for $v=4$ we have, for any $\omega\in\Omega_n^*$,
 $ |g_n(z)|\ge\frac12$.
By Lemma \ref{lem2.6}, we obtain inequality \eqref{mainbound} for $v=4$.
By Lemma \ref{crucial}, this inequality holds for any $v$ with $v_0 \le v\le 4$ as well.
Thus Proposition \ref{prop10} is proved.
 \end{proof}
\section{Proof of Theorem \ref{main}}To conclude the proof of Theorem \ref{main}
we shall now apply the result of Corollary \ref{smoothing1} with $v=\sqrt2 v_0$ and $V=4$ to the empirical spectral  distribution function
$\mathcal F_n(x)$ of the random matrix $\mathbf X$.
At first we bound the integral over the line  $V=4$. Note that in this case we have $|z+m_n(z)|\ge1$ and $|g_n(z)|\le \frac 12$ a.s.
Moreover, $\im m_n^{(j)}(z)\le \frac1V\le \frac12$. In this case the results of Lemmas \ref{deltan1}--\ref{deltan3} hold for any $z=u+4i$
with $u\in\mathbb R$. We   apply inequality \eqref{bound1}:
 \begin{align}\label{001} |g_n(z)|&\le\frac{\beta_n^{2}(1+\im^{\frac12}m_n(z)}{n\sqrt{v}|z+m_n(z)||z+s(z)+m_n(z)|}+\frac{ C\4  \im m_n(z)}
{n v|z+s(z)+m_n(z)|}\notag\\&+ \frac{ \beta_n^2}
{n^{\frac32} v|z+m_n(z)||z+s(z)+m_n(z)|}\notag\\&+
\frac{\beta_n^{4}}{nv|z+m_n(z)||z+s(z)+m_n(z)|}\Big(\im m_n(z)+\frac{1}{nv}\Big)\frac1n\sum_{j=1}^n|R_{jj}|,
\end{align}
which holds for any $z=u+4i$, $u\in\mathbb R$, with probability at least \newline$1-C\exp\{-cl_{n,\alpha}\}$.
Note that for $V=4$
\begin{equation} \notag
 |z+m_n(z)||z+m_n(z)+s(z)|\ge \begin{cases}4\quad\text{for}\quad |u|\le 2,\\
\frac14|z|^2\quad\text{for}|u|>2\end{cases}\text{ a. s.}
\end{equation}

We may rewrite the bound \eqref{001} as follows
\begin{align}
 |g_n(z)|\le\frac{C\beta_n^4}{n(|z|^2+1)}+\frac{ C\im m_n(z)}{nV}.\notag
\end{align}
Note that for any distribution function $F(x)$  we have
\begin{equation}\notag
 \int_{-\infty}^{\infty}\im s_F(u+iv)du\le \pi
\end{equation}
Moreover, for any random variable $\xi$ with distribution function $F(x)$ and $\E\xi=0$, $E\xi^2=h^2$ we have
\begin{equation}\notag
\im s_F(u+iV)\le\frac{C(1+h^2)}{u^2}
\end{equation}
with some numerical constant $C$.
From here it follows that, for $V=4$,
\begin{equation} \label{bigu}
\int_{|u|\ge n^2}|m_n(z)-s(z)|du\le \frac {C(1+h_n^2)}{n^2}\text{ a.s.}
\end{equation}
with $h_n^2=\int_{-\infty}^{\infty}x^2d\mathcal F_n(x)$.
Furthermore, note that
\begin{equation}\notag
 h_n^2=\frac1{n^2}\sum_{j,k=1}^nX_{jk}^2\le\frac2{n^2}\sum_{1\le j\le k\le n}X_{jk}^2.
\end{equation}
Using inequality \eqref{largedev}, we get
\begin{equation}\notag
 \Pr\{h_n^2>Cn\}\le C\exp\{-l_{n,\alpha}\}.
\end{equation}
The last inequality and inequality \eqref{bigu} together imply that
\begin{equation}\notag
 \int_{|u|>n^2}|m_n(u+iV)-s(u+iV)|du\le \frac Cn
\end{equation}
with probability at least $1-C\exp\{-c\4l_{n,\alpha}\}$.
Denote $\overline{\mathcal D}_n:=\{z=u+2i: |u|\le n^2\}$ and
$$
\overline \Omega_n:=\Bigl\{\cap_{z\in\overline{\mathcal D}_n}
\big\{\omega\in\Omega:\,|g_n(z)|\le\frac{C\beta_n^2}{n(|z|^2+1)}\big\}\Bigr\}\cap\Omega_n^*.$$
Using a  union bound, similar to \eqref{capprobab} we may show that
\begin{equation} \notag
 \Pr\{\overline \Omega_n\}\ge 1-C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
It is straightforward to check that for $\omega\in\overline \Omega_n$
\begin{equation}\label{daleko}
\int_{-\infty}^{\infty}|m_n(z)-s(z)|du\le \frac {C\beta_n^4}n.
\end{equation}
Furthermore, we put $\varepsilon=(2av_0)^{\frac23}$ and $v_0=\frac{d\beta_n^4}{n}$ with the constant $d$ as introduced in  \eqref{defd}.
To conclude the proof we need to consider the ''vertical'' integrals, for $z=x+iv'$ with $x\in\mathbb J'_{\varepsilon}$,
$v'=\frac{v_0}{\sqrt{\gamma}}$ and $\gamma=2-|x|$.
Note that
\begin{equation}\notag
 \int_{v'}^2\frac{\beta_n^4}{nv}dv\le \frac{C\beta_n^4\ln n}{n}.
\end{equation}
Furthermore,
\begin{equation} \notag
 \int_{v'}^2\frac1{n^2v^2\sqrt{\gamma+v}}dv\le \frac1{n^2v'\sqrt{\gamma}}\le \frac1{n^2v_0}\le \frac{\beta_n^4\ln n}{n}.
\end{equation}
Finally, we get, for any $\omega\in\overline\Omega_n$,
\begin{equation} \notag
 \Delta(\mathcal F_n,G)=\sup_x|\mathcal F_n(x)-G(x)|\le \frac {\beta_n^4\ln n}{n}.
\end{equation}
Thus Theorem \ref{main} is proved.
\vskip -0.5cm
\section{Proof of Theorem \ref{eigenvector}} We may express the  diagonal entries of the resolvent matrix $\mathbf R$ as follows
\begin{equation}\label{representeigenvector}
 R_{jj}=\sum_{k=1}^n\frac1{\lambda_k-z}|u_{jk}|^2.
\end{equation}
Consider the distribution function, say $F_{nj}(x)$, of the probability distribution of the eigenvalues  $\lambda_k$
\begin{equation} \notag
 F_{nj}(x)=\sum_{k=1}^n|u_{jk}|^2\mathbb I\{\lambda_k\le x\}.
\end{equation}
Then we have
\begin{equation} \notag
 R_{jj}= R_{jj}(z)=\int_{-\infty}^{\infty}\frac1{x-z}dF_{nj}(x),
\end{equation}
which means  that $R_{jj}$ is the Stieltjes transform of the distribution $F_{nj}(x)$.
Note that, for any $\lambda>0$,

\begin{equation} \notag
 \max_{1\le k\le n}|u_{jk}|^2\le \sup_x(F_{nj}(x+\lambda)-F_{nj}(x))=:Q_{nj}(\lambda).
\end{equation}
On the other hand side, it is easy to check that
 \begin{equation}\label{concentration}
 Q_{nj}(\lambda)\le 2\sup_u\lambda \4\im R_{jj}(u+i\lambda).
\end{equation}
By relations  \eqref{definition} and \eqref{capprobab}, we obtain, for any $v\ge v_0$ with $v_0=\frac{d\beta_n^4}{n}$ with a sufficiently large
constant $d$,
\begin{equation}\label{eigenvector1}
 \Pr\Bigl\{\frac{|\varepsilon_j|}{|z+m_n(z)|}\le \frac12 \Bigr\}\le C\exp\{-c \4 l_{n,\alpha}\}
\end{equation}
with constants $C$ and $c$ depending on $\varkappa$, $\alpha$ and $d$.
Furthermore, the representation \eqref{repr01} and inequality \eqref{eigenvector1} together imply, for $v\ge v_0$,\quad
$ \im R_{jj}\le |R_{jj}|\le C_1 $
with some positive constant $C_1>0$ depending on $\varkappa$ and $\alpha$.
This implies that
\begin{equation} \notag
 \Pr\Bigl\{\max_{1\le k\le n}|u_{jk}|^2\le \frac{\beta_n^4}{n}\Bigr\}\le C\exp\{-cl_{n,\alpha}\}.
\end{equation}
By a union bound we arrive at the inequality \eqref{deloc}.
To prove inequality \eqref{deloc1}, we consider the quantity
$ r_j:=R_{jj}-s(z)$.
Using equalities \eqref{repr01} and  \eqref{semicircular0}, we get
\begin{equation} \notag
 r_j=-\frac{s(z)g_n(z)}{z+m_n(z)}+\frac{\varepsilon_j}{z+m_n(z)}R_{jj}.
\end{equation}
By  inequalities \eqref{mainbound}, \eqref{important} and \eqref{capprobab}, we have
\begin{equation} \notag
 \Pr\{|r_j|\le \frac{c\beta_n^2}{\sqrt{nv}}\}\ge 1-C\exp\{-cl_{n,\alpha}\}.
\end{equation}
From here it follows that
\begin{equation} \notag
 \sup_{x\in\mathbb J_{\varepsilon}}\int_{v'}^V|r_j(x+iv)|dv\le \frac C{\sqrt n}.
\end{equation}
Similar to \eqref{daleko} we get
\begin{equation} \notag
 \int_{-\infty}^{\infty}|r_j(x+iV)|dx\le \frac {C\beta_n^2}{\sqrt n}.
\end{equation}
Applying Corollary \ref{smoothing1}, we get
\begin{equation} \notag
 \Pr\{\sup_x|F_{nj}(x)-G(x)|\le \frac{\beta_n^2}{\sqrt n}\}\ge 1-C\exp\{-c\4 l_{n,\alpha}\}.
\end{equation}
Using now that
\begin{equation} \notag
\Pr\Bigl\{\sup_x|\mathcal F_{n}(x)-G(x)|\le \frac{\beta_n^4\ln n}{ n}\Bigr\}\ge 1-C\exp\{-c\4 l_{n,\alpha}\},
\end{equation}
we get
\begin{equation} \notag
\Pr\Bigl\{\sup_x|F_{nj}(x)-\mathcal F_{n}(x)|\le \frac{\beta_n^2}{\sqrt n}\Bigr\}\ge 1-C\exp\{-c\4l_{n,\alpha}\}.
\end{equation}
Thus, Theorem \ref{eigenvector} is proved.

{\bf Acknowledgement.} The authors would like to thank S. G.  Bobkov for
drawing their attention to some references about large deviations for martingales and quadratic forms.

redpajama_set_name: RedPajamaArXiv


I've had my reasons to quit Basecamp, but I now I use Wrike [http://www.wrike.com] I like this service cause it's built just lik email. You can invite people and they can use their own accounts not just for collaboration with you, but also for managing their other projects. In fact that's how I got to know about the tool. My colleague sent me an invitation.
Maybe these are good online project management tools, but I can say for sure that ProjectOffice.net is more intuitive and easier to use. Why? ProjectOffice.net offers online project management, time management, issue tracking, wikis and much more, and the best thing is that it's free. So, why would you pay?
Thanks for the tip on Wrike. It certainly addresses one of my main complaints about Basecamp, which is the poor ease of use on my Blackberry. Email solves that, I'll certainly take a look at it.
ProjectOffice.net looks nice, but why do you say it is more intuitive and easier to use than Basecamp or Wrike? I'm always on the look out for better solutions so I wonder if it could fill my needs better?
I don't say ProjectOffice.net is the best PM solution there is, but it completely satisfies my PM needs. I don't need an advanced PM solution, although I need a little of everything, thus ProjectOffice.net is just right. It is more intuitive and easy to use because when you enter the application you know what to do, it has a natural workflow which is not quite the case with Wrike (I stopped using it because of the annoying callouts when you first register) and Basecamp (no user friendly GUI).
As for the OpenID, ProjectOffice.net doesn't support it, but you don't need, since you can work in several spaces with one login. You only choose the space from the dropdown list of spaces (accounts) you work in. There is no need for logging in and out.
The only thing that I am not particularly fond of is that ProjectOffice.net doesn't have reports. I hope they are considering it soon.
Thanks for the breakdown of ProjectOffice.net. I did check out their site and it for me, it just doesn't draw me away from Basecamp which works well for us. As I mentioned earlier @Fred the only thing Basecamp doesn't do well is the ability to reply to a message or post using email. I guess it's just different strokes for different folks – because I find the Basecamp GUI pretty easy to use.
Do work mostly on a Mac or a PC?
This is good info to have. I work for a fairly large/growing interactive company. One of my biggest complaints about basecamp first and foremost is their customer service. I have read their responses to other customers and I myself have received some pretty terrible responses. Either they all but tell you that you're stupid or they just tell you that it could never do that in the first place (when for the past couple years it did). That very large problem aside…now they've just become unreliable.
All in all it's just been one pain after another working with Basecamp. The many problems we've encountered are just too long to list. I'm at the point now, after several years of being a customer, that I'm going to terminate the account. I just have to find an alternative first, so this article and some of the responses are helpful because I hardly have the time to research another tool. I just wish that the service we currently use was reliable.


Mushroom wicker basket with shoulder strap and lock. Thanks to its dimension (33x27x32) and to its slightly rounded shape the basket has an excellent capacity. Moreover it is really light and you can use it in particular for mushroom picking or to collect any other kind of fruit and vegetable the the wood gives us!


Felipe VI is the King of Spain. He was born in 1968 and ascended the throne in June 2014, following the abdication of his father, King Juan Carlos I.
King Felipe is head of state and commander-in-chief of the Spanish Armed Forces - in accordance with the Spanish Constitution and role of the monarch.
Felipe's pre-throne title was 'Prince of Asturias' and he was working to support philanthropic causes and to promote international fellowship among Spanish-speaking countries.
King Felipe VІ is the world's third-youngest reigning monarch, after King Jigme Khesar Namgyel Wangchuck of Bhutan and Sheikh Tamim bin Hamad Al Thani of Qatar.
The King is married to Letizia Ortiz Rocasolano (Queen Letizia of Spain) - an award-winning television journalist formerly with CNN. Their engagement was announced on November 1, 2003 and they got married on May 22, 2004 in Madrid, Spain.
The King and The Queen always look extremely elegant and classy without being even a bit boring or too... conservative. Their ensembles always match perfectly, no matter if the King is wearing an uniform or a suit.
Felipe and Letizia have two daughters: Leonor, Princess of Asturias (2005) and Infanta Sofía (2007).
If you like King Felipe VI's style, support him in Be Global Fashion Network readers' voting for Most Stylish Men 2015.
Vote for King Felipe VI and win a made-to-measure jacket!


text:
On this date in 1979, Lene Lovich released her third single, "Bird Song". The song appeared on her second album, Flex. The B side was a non-LP track called "Trixi". This was added to Flex as a bonus track when it was re-released on CD. The 12 inch version had a third track, "Too Tender (To Touch)" which was from her debut album, Stateless.
Blogload: Lene Lovich - Flex - suckingicecubes.blogspot.com

redpajama_set_name: RedPajamaCommonCrawl


text:
NBA Stamp and Seal abolition: The unwanted brouhaha
By Temisan Amoye Sep 17, 2020
By Awa Kalu, SAN
There is a dark and ominous cloud hanging over the affairs of the Nigerian Bar Association which has led several eminent and "not so eminent" legal practitioners to offer explanations that either confound the situation or render some arguments quite inefficacious.
Prior to taking our substantive law courses in those days, the curriculum enabled us to have a bird's eye view of other fields such as Basic Logic, General Studies and Element of the Social Sciences.
We had a course called Use of English. Overall, even the process of socialization made it possible for altruism to be implanted in most young Lawyers as well as other students. I have not been a fan of the denigration of young Lawyers and I do not disdain what their generation stands for. What is surprising though is that what sociologists typify as generation gap is presently creating a real gap between the youthful bar and their older colleagues. What is more, there is now an appreciable disenchantment between the "SANs" and "non-SANs."
ALSO READ: It's confession hour in Abuja
The dichotomy became more manifest before, during and after the recently concluded elections into key offices across the hierarchy of the Nigerian Bar Association, NBA. Anyone in doubt ought to get affiliated to one or two platforms where Lawyers vent their spleen on topical issues. Some positions on the law or other national issues are contested on tendentious or spurious grounds- at times- devoid of logic or even common sense.
I was brought up in the tradition that required an argument to be marshalled in order to be understood then dissected and repudiated if need be but without the imputation of improper motive(s). These days, an argument is mutilated if you do not belong to the perceived ethnic group of the person whose contention is under scrutiny. The differences of opinion do not necessarily arise from ideology or the failure to persuade. You may simply be lampooned for one reason or another, not warranted by the topic or the opinion expressed.
I have taken the pain to offer the background above in order to explain the current brouhaha arising from the purported amendment of the protocol guiding the practice of Lawyers in this country. Prior to the instant 'amendment', what prevailed was a legal instrument made in 2007 and which remained in force until now.
The most prominent part of the instrument was the one that required a legal practitioner to affix his/her stamp or seal to a certain class of documents meant for the conduct of litigation and other sundry matters. The rule came to be contentious when some Lawyers sought to use it as a "trump card," a "destroyer", and the massive technicality to dislodge even a meritorious case.
The Supreme Court as the apex Court, brooking no nonsense, put a halt to the deployment of such boobytrap to obstruct the cause of justice. In Yaki&Anor V. Bagudu& others (2015) LPELR- 25721 (SC) the Supreme Court affirmed with finality that a document filed without a legal practitioner's stamp and seal is not thereby rendered null and void or even incompetent.
The failure occasions a mere irregularity capable of being remedied. That case has been followed in the other recent case of Wike V. Peterside where Kekere-Ekun J.S.C emphasized that, "with regard to the lack of NBA stamp and seal on the petition, I refer to the recent decision of this Court….. where this Court held that the failure to affix the approved seal ad stamp of the NBA on a process does not render the process null and void. It is an irregularity that can be cured by an application for extension of time and a deeming order".
ALSO READ: UNILAG Crisis: Visitation Panel submits report of findings to FG
Having regard to the decision of the Supreme Court indicating that heavy weather ought not be made of the failure to attach or affix the stamp and seal of a legal practitioner to Court processes or other transactional documents, can an amendment to the Rules of Professional Conduct by deleting this and kindred itemsbe a ground for the vehemence which it has occasioned?
I cannot personally fault the argument of those who have cited the relevant enabling instrument and came to the irrefutable conclusion that the Learned Attorney-General lacks the vires to alter our Rules of Professional Conduct all by himself. For the avoidance of doubt, it is within the Province of the General Council of the Bar to add or subtract from the Rules.
Consequently, those who have argued that the Learned Attorney-General is "right" because, he replicated what Chief Bayo Ojo, SAN, did in 2007 (by allegedly amending the Rules unilaterally) have missed the point. By the way, I am aware that a past Secretary-General of the NBA has affirmed that no such unilateral alteration took place. What ought to be done is for the NBA Secretariat (or the office of the Attorney-General) to publish the minutes of that meeting from the archives so that this matter may be put to bed, let Lawyers not hang themselves because of a matter which ought to be in record.
Furthermore, there are yet other members of the profession who have without proof, argued that the Learned Attorney-General simply wants to "assist" his "kinsmen" to entrench the "New Nigerian Bar Association"!Let it be made clear that what the Supreme Court has held time and again is that the Nigerian Bar Association is not a creature of statutes and thus, is not a juristic person.
This conclusion was first reached by the Supreme Court in 1989 in Fawehinmi V. NBA (No. 2) (1989) 2 NWLR (pt. 105). It was affirmed very recently in Moses Esq. V. Nigerian Bar Association (2019) LPELR- 46918 (SC). Can it then be doubted that an association formed by Lawyers does not need the imprimatur of the Attorney-General or his office to stand? Do we not at this time have the association exclusive to Muslim as well as Christian Lawyers? Do we not also have an Association of Female Lawyers just as Female Judges? In some states, we still have vestiges of "Judges Wives Association!" what about the Body of Senior Advocates of Nigeria (BOSAN)?
I know of families accommodating multiple wives and children including households which peacefully accommodate Christians, Muslims, Atheists etc, at the same time. The inevitable conclusion is that we as Lawyers, who are deemed to be Learned, and indeed Learned, have developed a penchant for obfuscating "threadbare matters" in the words of Professor Montrose.
None of us can deny that Lawyers have always stood for the defence of a right recognised and entrenched in the Constitution namely, Freedom of Association. Lawyers should also be able to imbibe the Mantra (live and lets live) and not succumb to Chinua Achebe's reference to the Falcon which cannot hear the Falconer, "Things Fall Apart", for that reason, the centre cannot hold.
The question at this point is whether we can tread softly on the destructive part that some of us have chosen in order to persuade rather than lampoon one another. There are larger associations of Lawyers than ours but we can argue vigorously, even doggedly but intelligently without being pedantic and obdurately dogmatic.
In the words of Senator Ben Bruce, "I just want to make common sense.
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Robert B. Shipp, 66, of Moline, passed away Sunday, February 10, 2019 at UnityPoint Health, Trinity-Rock Island. A memorial visitation will be held from 11 a.m. to 2 p.m. on Saturday at Wheelan Pressly Funeral Home and Crematory, 3030 7th Ave., Rock Island. Memorials may be made to METAvivor.
Bob was born August 7, 1952 in Amhurst, Ohio, son of Robert B. and Eleanor (Sharer) Shipp. He married Michele Rothweiler, his companion of 36+ years, on March 12, 2004. Though he held several jobs over the years, Bob's first love was cooking. He was a cook and sous chef at area restaurants and enjoyed cooking for family and friends. Bob was a licensed Pyrotechnician for 20 years and delighted in making memories for spectators. He was a lifelong camper and never met an animal or child who wasn't "taken" by his gentle way. He was very proud to be the first and only male model on the runway for "Stylin' against Breast Cancer." He was happy and proud to share his breast cancer journey. Bob was a 35 year member of the FOE #956, where he served as President for 11 years. Most of all, he was proud of his children and grandchildren whom he loved dearly. Special thanks to Dr. Stewart Garneau, his entire staff, and 4th floor south, Trinity, RI, for their care and compassion.
Survivors include his wife, Michele; parents, Robert and Eleanor; children, Corbin (Melissa) Shipp and Lacey (Robert) Burris; grandchildren, Hunter Shipp and Devin, Seth, Sarah and Benjamin Burris; siblings, Barry (Faye) Shipp, Rhonda Penn, Roy (Kathy) Shipp and many nieces, nephews, other loving family members and friends.
He was preceded in death by his sister, Frances and grandmother, Lizzie.