On Rozanov's Theorem and strenghtened asymptotic uniform distribution

TL;DR

This paper establishes conditions under which sums of independent integer-valued random variables exhibit asymptotic uniform distribution modulo h, strengthening Rozanov's theorem through local limit theorem assumptions and new probabilistic bounds.

Contribution

It provides a strengthened version of Rozanov's theorem linking local limit theorems to asymptotic uniform distribution of sums modulo h, with explicit bounds and relaxed conditions.

Findings

Proves a quantitative bound on the distribution of sums modulo h.

Shows that local limit theorem conditions imply asymptotic uniform distribution.

Provides strengthened forms of the asymptotic uniform distribution property.

Abstract

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 \phi(t)\uparrow \infty $ as $t\to \infty$, and some constant $C$, we have for all $n$ and $\nu\in \Z$, \begin{equation*}\label{abstract1} \big|B_n\P\big\{ S_n=\nu\big\}- {1\over \sqrt{ 2\pi } }\ e^{- {(\nu-M_n)^2\over 2 B_n^2} }\big|\,\le \, {C\over \,\phi(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{2\pi}\, B_n }+\frac{1+ 2 {C}/{h} }{ \phi(B_n)^{2/3} } + C_1 \,e^{-(1/ 16 )\phi(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 $\nu_n =\sum_{j=1}^n \t_j\uparrow \infty$. We prove also strenghtened forms of the asymptotic uniform distribution property.