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

TL;DR

This paper establishes uniform semi-local limit theorems for sums of i.i.d. lattice-valued random variables along sets of multiples, providing precise bounds and convergence rates for probabilities related to divisibility and residue classes.

Contribution

It introduces new uniform semi-local limit theorems along sets of multiples for sums of lattice-valued i.i.d. variables, with explicit error bounds and convergence rates.

Findings

Derived bounds for probability deviations in divisibility events

Established convergence rates for sums along residue classes

Provided uniform estimates over divisibility sets

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*}