Tightness of products of random matrices and stability of linear stochastic systems
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.
tightness of sequences of n-fold convolutions
scientific article; zbMATH DE number 3994626