Tightness of products of random matrices and stability of linear stochastic systems
type: Item
instance of: scholarly article
title: Tightness of products of random matrices and stability of linear stochastic systems
zbMATH Open document ID: 0614.60008
author: Philippe Bougerol
published in: The Annals of Probability
publication date: 1987-01-01T00:00:00Z
review text: The paper provides conditions implying tightness of sequences of n-fold convolutions \(\mu^ n\), \(n=1,2,..\). of probability distributions \(\mu\) on the set of real \(d\times d\)-matrices. These yield some applications to linear stochastic differential equations \[ (1)\quad dx_ t=S_ 0x_ tdt+\sum^{r}_{i=1}S_ ix_ t\circ db^ i_ t \] where \(S_ 0,...,S_ r\) are fixed \(d\times d\)-matrices and \(b^ i_ t\), \(i=1,...,r\) are independent Brownian motions. Namely, if the largest Lyapunov exponent of the above system of equations is zero then its zero solution is stable in probability if and only if there exists an invertible matrix Q such that for \(i=0,...,r\) \[ QS_ iQ^{-1}=\left( \begin{matrix} A_ i\\ 0\end{matrix} K_ i\begin{matrix} 0\\ B_ i\end{matrix} \right) \] where \(K_ i\) are skew-symmetric matrices and the largest Lyapunov exponents of systems of equations with the coefficients \(A_ 0,...,A_ r\) and \(B_ 0,...,B_ r\) in place of \(S_ 0,...,S_ r\) in (1) are strictly negative.
reviewed by: Yuri Kifer
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 93E15
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H25
zbMATH DE Number: 3994626
zbMATH Keywords: products of random matrices
zbMATH Keywords: stability of linear stochastic systems
zbMATH Keywords: tightness of sequences of n-fold convolutions
zbMATH Keywords: largest Lyapunov exponent
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176992256
OpenAlex ID: W2053093176
DOI: 10.1214/AOP/1176992256
description: scientific article; zbMATH DE number 3994626