Kolmogorov's law of the iterated logarithm for noncommutative martingales
type: Item
instance of: scholarly article
title: Kolmogorov's law of the iterated logarithm for noncommutative martingales
zbMATH Open document ID: 1335.46058
DOI: 10.1214/14-AIHP603
author: Qiang Zeng
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2015-10-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.1504
full work available at URL: https://projecteuclid.org/euclid.aihp/1435759242
review text: The law of the iterated logarithm is a main subject in probability theory. It is a strong almost sure theorem which gives a very precise bound on the almost sure fluctuations of the sums of random variables normalized as in the central limit theorem. The first studied case was for sums of independent identically distributed random variables, namely, Bernoulli random variables (\(\pm 1\) with \(\mathbb{P} = 1/2\)), this was done by Khintchine. This result was extended to independent but not identically distributed random variables by Kolmogorov when the variables are individually bounded but the bound grows at a controlled rate. \textit{P. Hartman} and \textit{A. Wintner} [Am. J. Math. 63, 169--176 (1941; Zbl 0024.15802)] proved that the finiteness of the second moment suffices for the law of the iterated logarithm in the independent identically distributed case. The case of martingales was first studied in 1937 by Lévy, later by \textit{V. Strassen} [in: Proc. 5th Berkeley Symp. Math. Stat. Probab., Univ. Calif. 1965/66, 2, Part 1, 315--343 (1967; Zbl 0201.49903)] and \textit{W. F. Stout} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 15, 279--290 (1970; Zbl 0209.49004)] under some boundedness conditions similar to those of Kolmogorov. In the present work, the author proves Kolomogorov's law of the iterated logarithm for noncommutative martingales.
reviewed by: Ghadir Sadeghi
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 6489497
zbMATH Keywords: law of iterated logarithm
zbMATH Keywords: noncommutative martingales
zbMATH Keywords: exponential inequality
MaRDI profile type: Publication
arXiv ID: 1212.1504
cites work: Q3998775
cites work: A new proof of the Hartman-Wintner law of the iterated logarithm
cites work: Generalized s-numbers of \(\tau\)-measurable operators
cites work: On the Law of the Iterated Logarithm
cites work: Doobs inequality for non-commutative martingales
cites work: Noncommutative Burkholder/Rosenthal inequalities
cites work: Noncommutative maximal ergodic theorems
cites work: Some mathematical models from population genetics. École d'Été de Probabilités de Saint-Flour XXXIX-2009
cites work: Q3997990
cites work: Q4215796
cites work: Q4440813
cites work: Q5611461
cites work: The Hartman-Wintner Law of the Iterated Logarithm for Martingales
cites work: Free Random Variables
cites work: Strong invariance principles for dependent random variables
cites work: Law of the iterated logarithm for stationary processes
description: scientific article