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

摘要

We determine the large exceedance probabilities and large exceedance pathsfor 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 byan application to multi-type branching processes. Other applications arise infinancial time series modeling (connected to the study of the GARCH($p,q$)processes) and in physics, and this recursive sequence has also been the focusof extensive work in the recent probability literature. In this work, wecharacterize 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 ${\mathbbR}^d$, showing that $T_u^A/u^\alpha$ converges to an exponential law. In theprocess, we also revisit and refine Kesten's classical estimate, showing thatif $V$ 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, mostimportantly, a new characterization of the constant $C_A$. Finally, we describethe large exceedance paths via two conditioned limit laws. In the first, weshow that conditioned on a large exceedance, the process $\{ V_n\}$ follows anexponentially-shifted Markov random walk, which we identify, therebygeneralizing 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 distributionconverges to the stationary law of the exponentially-shifted Markov randomwalk.