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 Vn=MnVn.1+Qn, where {(Mn,Qn)} is an i.i.d. sequence, and M1 is a d.d random matrix and Q1 is a random vector, both with nonnegative entries. We impose conditions which guarantee the existence of a unique stationary distribution for {Vn} and a Cramér-type condition for {Mn}. Under these assumptions, we characterize the distribution of the first passage time TAu:=inf{n:Vn.uA}, where A is a general subset of Rd, exhibiting that TAu/u. converges to an exponential law for a certain .>0. In the process, we revisit and refine classical estimates for P(V.uA), where V possesses the stationary law of {Vn}. Namely, for A.Rd, we show that P(V.uA).CAu.. as u.., providing, most importantly, a new characterization of the constant CA. As a simple consequence of these estimates, we also obtain an expression for the extremal index of {|Vn|}. Finally, we describe the large exceedance paths via two conditioned limit theorems showing, roughly, that {Vn} 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.