Abstract:

A sequence $X=\{X_i,i\ge 1\}$ of independent variables taking values in $\mathbb{Z}$, with partial sums $S_n=\sum _{k=1}^n{X}_k$, for each $n$ is said to be asymptotically uniformly distributed, or for short a.u.d., if:

$$\underset{n\to \mathrm{\infty }}{lim}\mathbb{P}\{S_n=m\text{(mod d)}\}=\frac{1}{d}$$

for any $d\ge 2$ and $m=0,1,\dots ,d-1$. We are concerned with estimating the rate of convergence in (0.1), a question recently discussed by Dolgopyat and Hafouta (2022). We obtain a quantitative estimate under weaker moment assumptions, by using a different approach.

We deduce from a more general result that if $(i)$ for some function $1\le \varphi (t)\uparrow \mathrm{\infty }$ as $t\to \mathrm{\infty }$, and some constant $C$, we have, for all $n$ and $\nu \in \mathbb{Z}$:

$$|B_n\mathbb{P}\{S_n=\nu \}-\frac{1}{\sqrt{2\pi }}{e}^{-\frac{(\nu -M_n{)}^2}{2B_n^2}}|\le \frac{C}{\varphi (B_n)},$$

then $(ii)$ there exists a numerical constant $C_1$ such that for all $n$ with $B_n\ge 6$, all $h\ge 2$, and $\mu =0,1,\dots ,h-1$:

$$|\mathbb{P}\{S_n\equiv \mu \text{(mod h)}\}-\frac{1}{h}|\le \frac{1}{\sqrt{2\pi B_n}}+\frac{1+2C/h}{\varphi (B_n{)}^{2/3}}+C_1{e}^{-\frac{1}{16}\varphi (B_n{)}^{2/3}}.$$

Assumption $(i)$ holds if a local limit theorem in the usual form is applicable, and $(ii)$ yields a strengthening of Rozanov's necessary condition.

Assume in place of $(i)$ that ${\vartheta }_j=\sum _{k\in \mathbb{Z}}\mathbb{P}\{X_j=k\}\wedge \mathbb{P}\{X_j=k+1\}>0$ for each $j$, and ${\nu }_n=\sum _{j=1}^n{\vartheta }_j\uparrow \mathrm{\infty }$. For these classes of random variables, we prove strengthened forms of the asymptotic uniform distribution property, with sharp uniform rate of convergence.

$(iii)$ Let $\alpha >{\alpha }^{\prime }>0$, $0<ϵ<1$. Then for each $n$ such that:

$$|x|\le \frac{1}{2}{(2\alpha \log ((1-ϵ){\nu }_n)\frac{1}{(1-ϵ){\nu }_n})}^{1/2}\Rightarrow \frac{\sin (x)}{x}\ge \sqrt{1-ϵ},$$

we have:

$$\underset{u\ge 0}{sup}\underset{d<\pi {\left(\frac{(1-ϵ){\nu }_n}{2\alpha \log ((1-ϵ){\nu }_n)}\right)}^{1/2}}{sup}|\mathbb{P}\{d\mid S_n+u\}-\frac{1}{d}|\le 2e^{-\frac{{ϵ}^2{\nu }_n}{2}}+{((1-ϵ){\nu }_n)}^{-{\alpha }^{\prime }}.$$

2010 AMS Mathematics Subject Classification: Primary 60F15; Secondary 60G50, 60F05.

Keywords and phrases: local limit theorem, asymptotic uniform distribution, independent sequences of random variables, Rozanov's theorem, Poisson summation formula, divisors, Bernoulli random variables, elliptic Theta functions.