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# A Berry--Esseen theorem for weakly negatively dependent random variables and its applications

Acta Mathematica Hungarica
Authors: Jianfeng Wang and Lixin Zhang

## Summary

We provide uniform rates of convergence in the central limit theorem for linear negative quadrant dependent (LNQD) random variables. Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\{X_{n},\allowbreak n\ge1\}$ \end{document} be a LNQD sequence of random variables with \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $EX_{n}=0$ \end{document}, set \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $S_{n}=\sum_{j=1}^{n}X_{j}$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $B_{n}^{2}=\text{Var}\, (S_{n})$ \end{document}. We show that \begin{gather*} \sup_{x} \left|P\left(\frac{S_{n}}{B_{n}}<x\right)-\Phi(x)\right|= O\bigg(n^{-\delta/(2+3\delta)}\vee \frac{n^{3\delta^{2}/(4+6\delta)}}{B^{2+\delta}_{n}} \sum_{i=1}^{n} E{|X_{i}|}^{2+\delta}\bigg) \end{gather*} under finite \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(2+\delta)$ \end{document}th moment and a power decay rate of covariances. Moreover, by the truncation method, we obtain a Berry--Esseen type estimate for negatively associated (NA) random variables with only finite second moment. As applications, we obtain another convergence rate result in the central limit theorem and precise asymptotics in the law of the iterated logarithm for NA sequences, and also for LNQD sequences.

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# Strong approximation for sums of asymptotically negatively dependent Gaussian sequences with applications

Acta Mathematica Hungarica
Authors: K.-A. Fu and L.-X. Zhang

## Abstract

By applying the Skorohod martingale embedding method, a strong approximation theorem for partial sums of asymptotically negatively dependent (AND) Gaussian sequences, under polynomial decay rates, is established. As applications, the law of the iterated logarithm, the Chung-type law of the iterated logarithm and the almost sure central limit theorem for AND Gaussian sequences are derived.

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# LIL behavior for weakly dependent random variables in Banach spaces

Acta Mathematica Hungarica
Author: K. Fu

## Abstract

Given a sequence of identically distributed ψ-mixing random variables {X n; n ≧ 1} with values in a type 2 Banach space B, under certain conditions, the law of the iterated logarithm for this sequence is obtained without second moment.

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# A Probability Limit Theorem for {f(nx)}

Acta Mathematica Hungarica
Author: Erika Péter

## Abstract

We prove a law of the iterated logarithm for sums ∑kN f(n k x) where f is a periodic measurable function and (n k) is a rapidly increasing sequence of integers. Our result applies also in the sub-Hadamard case and extends and improves earlier results in the field.

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# A law of the single logarithm for moving averages of random variables under exponential moment conditions

Studia Scientiarum Mathematicarum Hungarica
Author: H. Lanzinger

GAPOSKIN, V. F., The law of the iterated logarithm for Abel's and Cesàro's methods of summation, Teor. Verojatnost. i Primenen . 10 (1965), 449-459 (in Russian). MR 33 #3336 The law

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# Functional limit theorems for digital expansions

Acta Mathematica Hungarica
Authors: M. Drmota, M. Fuchs, and E. Manstavičius

## Abstract

The main purpose of this paper is to discuss the asymptotic behaviour of the difference s q,k(P(n)) - k(q-1)/2 where s q,k (n) denotes the sum of the first k digits in the q-ary digital expansion of n and P(x) is an integer polynomial. We prove that this difference can be approximated by a Brownian motion and obtain under special assumptions on P, a Strassen type version of the law of the iterated logarithm. Furthermore, we extend these results to the joint distribution of q 1-ary and q 2-ary digital expansions where q 1 and q 2 are coprime.

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# A nonclassical LIL for sums of B-valued random variables when extreme terms are excluded

Acta Mathematica Hungarica
Author: Ke-Ang Fu

10474-008-7236-1 .  Klesov , O. Rosalsky , A. 2001 A nonclassical law of the iterated logarithm for i.i.d. square integrable random

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# Small ball estimates for Brownian motion under a weighted sup-norm

Studia Scientiarum Mathematicarum Hungarica
Authors: Ph. Berthet and Z. Shi

iterated logarithm, Z. Wahrsch. Verw. Gebiete 54 (1980), 287-301. MR 82d :60052 A relation between Chung's and Strassen's laws of the iterated logarithm Z. Wahrsch. Verw. Gebiete

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# Gap series and functions of bounded variation

Acta Mathematica Hungarica
Author: Katusi Fukuyama

## Summary

We prove that the gap seriesΣf(n k x) does not behave like an independent random series when f is a function of bounded variation with rational discontinuity.

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# AN EXTENSION OF THE SUBSE UENCE PRINCIPLE

Studia Scientiarum Mathematicarum Hungarica
Author: E. Péter

subsequence principle in probability theory. II. The law of the iterated logarithm, Invent. Math. 25 (1974), 241-251. MR 50 #11403 A subsequence principle in probability theory. II. The law of the iterated logarithm, Invent

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