Almost sure limit theorems for random sums of multiindex random variables
type: Item
instance of: scholarly article
title: Almost sure limit theorems for random sums of multiindex random variables
zbMATH Open document ID: 1247.60042
author: L. P. Terekhova
published in: Lithuanian Mathematical Journal
publication date: 2009-11-06T00:00:00Z
review text: Denote by \(\mathop \to \limits^d \) the convergence in distribution, by \(\mathop \to \limits^p \) the convergence in probability, by \(\mathop \to \limits^w \) the weak convergence of measures, and by \({\mu _\varsigma }\) the distribution of the random variable \(\varsigma \). Let \({\varsigma _n}\) be a sequence of random variables defined on the probability space \((\Omega ,A,P)\). Consider the measures \({Q_n}(\omega ) = {Q_n}(({\varsigma _n}))(\omega ) = \frac{1}{{\log n}}\sum_{k = 1}^n {\frac{1}{k}{\delta _{{\varsigma _k}(\omega )}}} \), where \(\omega \in \Omega \), and \({\delta _x}\) is the point mass at \(x\). Classical limit theorems deal with the convergence \({\varsigma _n}\mathop \to \limits^d \varsigma \) as \(n \to \infty \). In many cases, this convergence implies the convergence of measures \({Q_n}(\omega )\mathop \to \limits^w {\mu _\varsigma }\) as \(n \to \infty \) for almost all \(\omega \in \Omega \). Such limit theorems are called almost sure versions of ordinary limit theorems. In this paper, the author obtains a functional limit theorem for multiindex random sums and its almost sure version.
reviewed by: Oleg K. Zakusilo
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 5626717
zbMATH Keywords: almost sure limit theorems
zbMATH Keywords: functional limit theorems
zbMATH Keywords: random sums
zbMATH Keywords: multiindex random variables
MaRDI profile type: Publication
full work available at URL: s10986-009-9051-y
OpenAlex ID: W2052616554
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DOI: 10.1007/S10986-009-9051-Y
description: scientific article; zbMATH DE number 5626717