LARGE EXCURSIONS AND CONDITIONED LAWS FOR RECURSIVE SEQUENCES GENERATED BY RANDOM MATRICES

We study the large exceedance probabilities and large exceedance paths of the recursive sequence V-n = MnVn-1 + Q(n), where {(M-n, Q(n))} is an i.i.d. sequence, and M-1 is a d x d random matrix and Q(1) is a random vector, both with nonnegative entries. We impose conditions which guarantee the existence of a unique stationary distribution for {V-n} and a Cramr-type condition for {M-n}. Under these assumptions, we characterize the distribution of the first passage time T-u(A) = inf{n : V-n is an element of uA}, where A is a general subset of R-d, exhibiting that T-u(A)/u(alpha) converges to an exponential law for a certain alpha > 0. In the process, we revisit and refine classical estimates for P(V is an element of uA), where V possesses the stationary law of {V-n}. Namely, for A subset of R-d, we show that P(V is an element of uA) similar to C-A(u-alpha) as V -> infinity, providing, most importantly, a new characterization of the constant C-Lambda. As a simple consequence of these estimates, we also obtain an expression for the extremal index of {vertical bar V-n vertical bar}. Finally, we describe the large exceedance paths via two conditioned limit theorems showing, roughly, that {V-n} follows an exponentially-shifted Markov random walk, which we identify. We thereby generalize results from the theory of classical random walk to multivariate recursive sequences.