Distribution of a random power series
type: Item
instance of: scholarly article
title: Distribution of a random power series
zbMATH Open document ID: 0576.60017
DOI: 10.1007/BF00970330
published in: Lithuanian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: In this note the author has generalized the result of \textit{R. Ranga Rao} [Ann. Math. Stat. 33, 659--680 (1962; Zbl 0117.28602)] to the two-dimensional case. Let \(\eta_ i\), \(i=1,2,..\). be independent random vectors of the space \(\mathbb R^ 2\) with identical non-degenerate distribution and let the series \[ \psi =\eta_ 1+\eta_ 2t+\eta_ 3t^ 2+...+\eta_ nt^{n-1}+... \tag{*} \] converge, \(t\in (-1,0)\cup (0,1)\). Then for the boundary \(\Gamma\) of any convex set \(P(\sum^{\infty}_{i=1}\eta_ it^{i-1}\in \Gamma)=0\). He has also proved that the distributions of partial sums of the series (*) converge uniformly on all Borel convex sets in \(\mathbb R^ 2\) if and only if for any point \(x\in \mathbb R^ 2-\{0\}\), the probability \(P(\eta_ i=x)<1\).
Mathematics Subject Classification ID: 60E99
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 3921612
zbMATH Keywords: random power series
zbMATH Keywords: convex sets
zbMATH Keywords: uniform convergence
author: A. K. Grincevičius
reviewed by: V. Thangaraj
MaRDI profile type: Publication
cites work: Random geometric series and intersymbol interference
cites work: Stochastic groups
cites work: Nouvelles classes de lois limites
cites work: Relations between Weak and Uniform Convergence of Measures with Applications
cites work: A Generalized Bernoulli Scheme and Its Limit Distributions
full work available at URL: https://doi.org/10.1007/bf00970330
OpenAlex ID: W2010079429
description: scientific article; zbMATH DE number 3921612