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

Abstract

Let $X$ be a square integrable random variable with basic probability space $(Ø, \A, ¶)$, taking values in a lattice $\mathcal{L}(v_0,1)=\{v_k=v_0+k,k\in \mathbb{Z}\}$ 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 $S_n={\sum }_{j=1}^n{X}_j$, 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 $\rho$ 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 $F_d=d\mathbb{N}$. \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*}

20 pages

Cited by in corpus (1)

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