On a law of the iterated logarithm for sums mod 1 with application to Benford's law
type: Item
instance of: scholarly article
title: On a law of the iterated logarithm for sums mod 1 with application to Benford's law
zbMATH Open document ID: 0619.60032
DOI: 10.1007/BF00334035
author: Peter Schatte
publication date: 1988-01-01T00:00:00Z
review text: Let \(Z_ n\) be the sum mod 1 of n i.i.d. r.v. and let \(1_{[0,x)}(\cdot)\) be the indicator function of the interval [0,x). Then the sequence \(1_{[0,x)}(Z_ n)\) does not converge for any x. But if arithmetic means are applied then under suitable suppositions convergence with probability one is obtained for all x as is well-known. In the present paper the rate of this convergence is shown to be of order \(n^{-1/2}(\log \log n)^{1/2}\) by using estimates of the remainder term in the CLT for m-dependent r.v.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 4005244
zbMATH Keywords: law of the iterated logarithm
zbMATH Keywords: central limit theorem
zbMATH Keywords: rate of convergence
MaRDI profile type: Publication
cites work: On roundoff error distributions in floating point and logarithmic arithmetic
cites work: Q5768351
cites work: Speed of convergence of the n-fold convolution of a probability measure on a compact group
cites work: Q4114551
cites work: Q5615145
cites work: A method for the derivation of limit theorems for sums of m-dependent random variables
cites work: Konvergenzgeschwindigkeit der Folge der Faltungspotenzen eines Wahr‐scheinlichkeitsmaßes auf einer kompakten topologischen Gruppe
cites work: On the uniform distribution of sequences of random variables
cites work: Q4767350
cites work: Some results in the probabilistic theory of asymptotic uniform distribution modulo 1
cites work: Q4085018
cites work: A functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables
cites work: The First Digit Problem
cites work: On the Equidistribution of Sums of Independent Random Variables
cites work: Zur Verteilung der Mantisse in der Gleitkommadarstellung einer Zufallsgröße
cites work: On the Asymptotic Uniform Distribution of Sums Reduced mod 1
cites work: The asymptotic uniform distribution modulo 1 of cumulative processes
cites work: On the Asymptotic Uniform Distribution of then-Fold Convolution mod 1 of a Lattice Distribution
cites work: Q3777153
cites work: On the Almost Sure Convergence of Floating-Point Mantissas and Benford's Law
cites work: Q3813201
cites work: On the asymptotic uniform distribution of stochastic clearing processes
cites work: Q3878450
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4005244