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 $(\O, \A, \P)$, taking values in a lattice $\mathcal L(v_0,1)=\big\{v_k=v_0+ k,k\in \Z\big\}$ and such that $\t_X =\sum_{k\in \Z}\P\{X=v_k\}\wedge \P\{X=v_{k+1}\}>0$. 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 $\t=1-\m$, $s(t) =\sum_{k\in \Z} \m_k\, e^{ 2i \pi v_kt}$ and $\rho$ be such that $1-\t<\rho<1$. 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 $\theta=\theta(\rho,\t)$ with $ 0< \theta <\t$, $C$ and $N$ such that for $ n \ge N$, \begin{align*} \sup_{u\ge 0}\,\sup_{d\ge 2} \Big| \P \{ S_n+u\in F_{d} \} - {1\over d}- {1\over d}\sum_{ 0< |\ell|<d }& \Big( e^{ (i\pi {\ell\over d }-{ \pi^2\ell^2\over 2 d^2}) } \t\, \E \,e^{2i \pi {\ell\over d }\widetilde X } +s\big( {\ell\over d }\big)\Big)^n \Big| \cr &\le \frac{C }{ \theta ^{3/2}}\ \frac{(\log n)^{5/2}}{ n^{3/2}}+2\rho^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| \P \{d|S_n+u \} - {1\over d}\Big| & \le & \frac{C_1}{\t} \, + \frac{C_2 }{ \theta ^{3/2}} +\frac{2\rho^2}{1-\rho}. \end{eqnarray*}