Large excursions and conditioned laws for recursive sequences generated by random matrices

We determine the large exceedance probabilities and large exceedance paths for the matrix recursive sequence \(V_n = M_n V_{n-1} + Q_n, \: n=1,2,\ldots,\) where \(\{M_n\}\) is an i.i.d. sequence of \(d \times d\) random matrices and \(\{ Q_n\}\) is an i.i.d. sequence of random vectors, both with nonnegative entries. Early work on this problem dates to Kesten's (1973) seminal paper, motivated by an application to multi-type branching processes. Other applications arise in financial time series modeling (connected to the study of the GARCH(\(p,q\)) processes) and in physics, and this recursive sequence has also been the focus of extensive work in the recent probability literature. In this work, we characterize the distribution of the first passage time \(T_u^A := \inf \{n: V_n \in u A \}\), where \(A\) is a subset of the nonnegative quadrant in \({\mathbb R}^d\), showing that \(T_u^A/u^\alpha\) converges to an exponential law. In the process, we also revisit and refine Kesten's classical estimate, showing that if \(V\) has the stationary distribution of \(\{ V_n \}\), then \({\mathbb P} \left( V \in uA \right) \sim C_A u^{-\alpha}\) as \(u \to \infty\), providing, most importantly, a new characterization of the constant \(C_A\). Finally, we describe the large exceedance paths via two conditioned limit laws. In the first, we show that conditioned on a large exceedance, the process \(\{ V_n\}\) follows an exponentially-shifted Markov random walk, which we identify, thereby generalizing results for classical random walk to matrix recursive sequences. In the second, we characterize the empirical distribution of \(\{ \log |V_n| - \log |V_{n-1}| \}\) prior to a large exceedance, showing that this distribution converges to the stationary law of the exponentially-shifted Markov random walk.