Tightness of products of random matrices and stability of linear stochastic systems. (English) Zbl 0614.60008

Bougerol, Philippe

Ann. Probab. 15, 40-74 (1987).

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.

Reviewer: Y.Kifer

MSC:

60B15	Probability measures on groups or semigroups, Fourier transforms, factorization
60B10	Convergence of probability measures
93E15	Stochastic stability in control theory
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H25	Random operators and equations (aspects of stochastic analysis)

Keywords:

products of random matrices; stability of linear stochastic systems; tightness of sequences of n-fold convolutions; largest Lyapunov exponent