Asymptotic behaviour under iterated random linear transformations
type: Item
instance of: scholarly article
title: Asymptotic behaviour under iterated random linear transformations
zbMATH Open document ID: 1055.43002
author: Riddhi Shah
author: S. G. Dani
published in: Mathematical Research Letters
publication date: 2004-11-01T00:00:00Z
review text: Let \(X_1,X_2,\dots\) be independent random variables with values in the space \(GL(V)\) of linear transformations of a finite dimensional real vector space \(V\) with identical distributions \(\mu\). Then the product \(Z_i=X_i\cdot X_{i-1}\dots X_1\) corresponds to the application of \(i\) random linear transformations. Let \(G(\mu)\) denote the smallest closed subgroup of \(GL(V)\) containing the support of \(\mu\). The authors study the limit behaviour of \(Z_i\cdot\nu\) when \(\nu\) is a probability measure on \(V\). Let \(B\) be the subspace of \(V\) containing all elements \(v\) such that the \(G(\mu)\)-orbit of \(v\) is bounded in \(V\). Among other results, it is shown that \(Z_i\cdot\nu\) tends to 0 in the vague topology if and only if \(\nu(B)=0\).
reviewed by: Ulrich Krengel
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 60B15
zbMATH DE Number: 2113043
MaRDI profile type: Publication
full work available at URL: mrl.2004.v11.n4.a6
OpenAlex ID: W1978054313
DOI: 10.4310/MRL.2004.V11.N4.A6
description: scientific article; zbMATH DE number 2113043