Abstract. The purpose of this paper is to extend a recent result on the Lyapunov-exponent of a stationary, ergodic sequence of block-triangular random matrices to the problem of $L_q$-stability for i.i.d. sequences of block-triangular random matrices. A known sufficient condition for $L_q$-stability of an i.i.d. sequence of random matrices $A_n$, with $q$ even, is that $\rho[ {\rm E}(A^{\otimes q}) ] < 1$, where $\rho $ is the spectral radius. It is shown that the validity of this condition for the diagonal blocks of $A$ implies its validity for the full matrix, see Theorem 1.1. A brief survey of results on $L_q$-stability, and a simple proof of the above sufficient condition will be given. Two major areas of applications, modelling and estimation of bilinear time series and stochastic volatility processes will be also briefly described.

AMS Subject Classification (1991): 93E15, 34D08

Keyword(s): random matrix products, Lyapunov exponents, higher order moments, bilinear models, GARCH processes

Received July 24, 2007, and in final form June 19, 2008. (Registered under 6054/2009.)