Abstract:

A sequence \( X = \{X_i, i \geq 1\} \) 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:

\[ \lim_{n \to \infty} \mathbb{P}\{S_n = m \, \text{(mod d)}\} = \frac{1}{d} \]

for any \( d \geq 2 \) and \( m = 0, 1, \ldots, d - 1 \). 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 \( 1 \leq \phi(t) \uparrow \infty \) as \( t \rightarrow \infty \), and some constant \( C \), we have, for all \( n \) and \( \nu \in \mathbb{Z} \):

\[ \left| B_n \mathbb{P}\{S_n = \nu\} - \frac{1}{\sqrt{2\pi}} e^{-\frac{(\nu - M_n)^2}{2B_n^2}} \right| \leq \frac{C}{\phi(B_n)}, \]

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

\[ \left| \mathbb{P}\{S_n \equiv \mu \, \text{(mod h)}\} - \frac{1}{h} \right| \leq \frac{1}{\sqrt{2\pi B_n}} + \frac{1 + 2C/h}{\phi(B_n)^{2/3}} + C_1e^{-\frac{1}{16}\phi(B_n)^{2/3}}. \]

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 \( \vartheta_j = \sum_{k \in \mathbb{Z}} \mathbb{P}\{X_j = k\} \land \mathbb{P}\{X_j = k + 1\} > 0 \) for each \( j \), and \( \nu_n = \sum_{j=1}^{n} \vartheta_j \uparrow \infty \). For these classes of random variables, we prove strengthened forms of the asymptotic uniform distribution property, with sharp uniform rate of convergence.

\( (iii) \) Let \( \alpha > \alpha' > 0 \), \( 0 < \epsilon < 1 \). Then for each \( n \) such that:

\[ |x| \leq \frac{1}{2} \left( 2\alpha \log((1 - \epsilon) \nu_n) \frac{1}{(1 - \epsilon) \nu_n} \right)^{1/2} \Rightarrow \frac{\sin(x)}{x} \geq \sqrt{1 - \epsilon}, \]

we have:

\[ \sup_{u \geq 0} \sup_{d < \pi \left( \frac{(1 - \epsilon) \nu_n}{2 \alpha \log((1 - \epsilon) \nu_n)} \right)^{1/2}} \left| \mathbb{P}\{d \mid S_n + u\} - \frac{1}{d} \right| \leq 2e^{-\frac{\epsilon^2 \nu_n}{2}} + \left((1 - \epsilon) \nu_n\right)^{-\alpha'}. \]

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.