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Michel J. G. Weber

Publications and source records attributed to Michel J. G. Weber.

6 recordsLinked to original sources
arXiv2025

A general correlation inequality for level sets of sums of independent random variables using the Bernoulli part with applications to the almost sure local limit theorem

Let $X=\{X_j , j\ge 1\}$ be a sequence of independent, square integrable variables taking values in a common lattice $\mathcal L(v_{ 0},D )= \{v_{ k}=v_{ 0}+D k , k\in \Z\}$. Let $S_n=X_1+\ldots +X_n$, $a_n= {\mathbb E\,} S_n$, and $\s_n^2={\rm Var}(S_n)\to \infty$ with $n$. Assume that for each $j$, $\t_{X_j} =\sum_{k\in \Z}{\mathbb P}\{X_j=v_k\}\wedge{\mathbb P}\{X_j=v_{k+1}\}>0$. Using the Bernoulli part, we prove a general sharp correlation inequality extending the one we obtained in the i.i.d.\,case in \cite{W3}: Let $0<\t_j\le \t_{X_j}$ and assume that $ ν_n =\sum_{j=1}^n \t_j \, \uparrow \infty$, $n\to \infty$. Let $\k_j\in \mathcal L(jv_0,D)$, $j=1,2,\ldots$ be a sequence of integers such that \begin{equation*} {\rm(1)}\qquad\frac{κ_j-a_j}{\s_j}=\mathcal O(1 ), \qq\quad {\rm(2)}\qquad \s_j \,{\mathbb P}\{S_j=κ_j\} ={\mathcal O}(1). \end{equation*} Then there exists a constant $C $ such that for all $1\le m<n$, \begin{align*} \s_n&\s_m \, \Big|{\mathbb P}\{S_n=\k_n, S_m=\k_m\}- {\mathbb P}\{S_n=\k_n \}{\mathbb P}\{ S_m=\k_m\} \Big| \cr & \,\le \, \frac{C}{D^2}\, \max \Big(\frac{\s_n }{\sqrt{ν_n}},\frac{\s_m }{\sqrt {ν_m}} \Big)^3 \,\bigg\{ ν_n^{1/2} \prod_{j=m+1}^n\vartheta_j + {ν_n^{1/2} \over (ν_n-ν_m) ^{3/2}}+{ 1\over \sqrt{ν_n\over ν_m}-1} \bigg\}. \end{align*} We derive a sharp almost sure local limit theorem

math.PR↗
arXiv2024

Divisibility and primality in random walks

In this paper we study the divisibility and primality properties of the Bernoulli random walk. We improve or extend some of our divisibility results to wide classes of iid or independent non iid random walks. We also obtain new primality results for the Rademacher random walk. We study the value distribution of divisors of the random walk in the Cramér model, and obtain a general estimate of a similar kind to that of the Bernouilli model. Earlier results on divisors and quasi-prime numbers in the Bernoulli model are recorded, as well as some other recent for the Cramér random model, based on an estimate due to Selberg.

math.PR↗
arXiv2022

On Rozanov's Theorem and strenghtened asymptotic uniform distribution

For sums $S_n=\sum_{k=1}^n X_k$, $n\ge 1$ of independent random variables $ X_k $ taking values in $\Z$ we prove, as a consequence of a more general result, that if (i) For some function $1\le ϕ(t)\uparrow \infty $ as $t\to \infty$, and some constant $C$, we have for all $n$ and $ν\in \Z$, \begin{equation*}\label{abstract1} \big|B_n¶\big\{ S_n=ν\big\}- {1\over \sqrt{ 2π} }\ e^{- {(ν-M_n)^2\over 2 B_n^2} }\big|\,\le \, {C\over \,ϕ(B_n)}, \end{equation*} then (ii) There exists a numerical constant $C_1$, such that for all $n $ such that $B_n\ge 6$, all $h\ge 2$, and $\m=0,1,\ldots, h-1$, \begin{align*}\label{abstract1} \Big|{\mathbb P}\big\{ S_n\equiv\, \m\ \hbox{\rm{ (mod $h$)}}\big\}- \frac{1}{h}\Big| \le {1\over \sqrt{2π}\, B_n }+\frac{1+ 2 {C}/{h} }{ ϕ(B_n)^{2/3} } + C_1 \,e^{-(1/ 16 )ϕ(B_n)^{2/3}}. \end{align*} Assumption (i) holds if a local limit theorem in the usual form is applicable, and (ii) yields a strenghtening of Rozanov's necessary condition. Assume in place of (i) that $\t_j =\sum_{k\in \Z}{\mathbb P}\{X_j= k\}\wedge{\mathbb P}\{X_j= k+1 \} >0$, for each $j$ and that $ν_n =\sum_{j=1}^n \t_j\uparrow \infty$. We prove also strenghtened forms of the asymptotic uniform distribution property.

math.PR↗
arXiv2022

A uniform semi-local limit theorem along sets of multiples for sums of i.i.d. random variables

Let $X $ be a square integrable random variable with basic probability space $(Ø, \A, ¶)$, taking values in a lattice $\mathcal L(v_0,1)=\big\{v_k=v_0+ k,k\in \Z\big\}$ and such that $\t_X =\sum_{k\in \Z}¶\{X=v_k\}\wedge ¶\{X=v_{k+1}\}>0$. Let $ X_i$, $i\ge 1 $ be independent, identically distributed random variables having same law than $X$, and let $S_n=\sum_{j=1}^nX_j$, for each $n$. Let $\m_k\ge 0$ be such that $ \m= \sum_{k\in \Z}\m_k $ verifies $1- \t_X<\m<1$, noting that $\t_X< 1$ always. Further let $\t=1-\m$, $s(t) =\sum_{k\in \Z} \m_k\, e^{ 2i πv_kt}$ and $ρ$ be such that $1-\t<ρ<1$. We prove the following uniform semi-local theorems for the class $\mathcal F=\{F_{d}, d\ge 2\}$, where $F_{d}= d\N$. \noi(i) There exists $θ=θ(ρ,\t)$ with $ 0< θ<\t$, $C$ and $N$ such that for $ n \ge N$, \begin{align*} \sup_{u\ge 0}\,\sup_{d\ge 2} \Big| ¶\{ S_n+u\in F_{d} \} - {1\over d}- {1\over d}\sum_{ 0< |\ell|<d }& \Big( e^{ (iπ{\ell\over d }-{ π^2\ell^2\over 2 d^2}) } \t\, \E \,e^{2i π{\ell\over d }\widetilde X } +s\big( {\ell\over d }\big)\Big)^n \Big| \cr &\le \frac{C }{ θ^{3/2}}\ \frac{(\log n)^{5/2}}{ n^{3/2}}+2ρ^n. \end{align*} \vskip 1 pt \noi(ii) Let $\mathcal D$ be a test set of divisors $\ge 2$, $\mathcal D_\p$ be the section of $\mathcal D$ at height $\p$ and $|\mathcal D_\p|$ denote its cardinality. Then, \begin{eqnarray*} \sum_{n=N}^\infty \ \sup_{u\ge 0} \, \sup_{\p\ge 2}\, {1\over |\mathcal D_\p |} \sum_{d\in \mathcal D_\p } \,\Big| ¶\{d|S_n+u \} - {1\over d}\Big| & \le & \frac{C_1}{\t} \, + \frac{C_2 }{ θ^{3/2}} +\frac{2ρ^2}{1-ρ}. \end{eqnarray*}

math.PR↗
arXiv2011

On small deviations of stationary Gaussian processes and related analytic inequalities

Let $ \{X_j, j\in \Z\}$ be a Gaussian stationary sequence having a spectral function $F$ of infinite type. Then for all $n$ and $z\ge 0$,$$ ¶\Big\{\sup_{j=1}^n |X_j|\le z \Big\}\le \Big(\int_{-z/\sqrt{G(f)}}^{z/\sqrt{G(f)}} e^{-x^2/2}\frac{\dd x}{\sqrt{2π}} \Big)^n,$$ where $ G(f)$ is the geometric mean of the Radon Nycodim derivative of the absolutely continuous part $f$ of $F$. The proof uses properties of finite Toeplitz forms. Let $ \{X(t), t\in \R\}$ be a sample continuous stationary Gaussian process with covariance function $\g(u) $. We also show that there exists an absolute constant $K$ such that for all $T>0$, $a>0$ with $T\ge \e(a)$, $$¶\Big\{\sup_{0\le s,t\le T} |X(s)-X(t)|\le a\Big\} \le \exp \Big \{-{KT \over \e(a) p(\e(a))}\Big\} ,$$ where $\e (a)= \min\big\{b>0: \d (b)\ge a\big\}$, $\d (b)=\min_{u\ge 1}\{\sqrt{2(1-\g((ub))}, u\ge 1\}$, and $ p(b) = 1+\sum_{j=2}^\infty {|2\g (jb)-\g ((j-1)b)-\g ((j+1)b)| \over 2(1-\g(b))}$. The proof is based on some decoupling inequalities arising from Brascamp-Lieb inequality. Both approaches are developed and compared on examples. Several other related results are established.

math.PR↗