Limit theorems for first passage times of multivariate perpetuity sequences

Abstract

We study the first passage time $\tau_u = \inf \{ n \geq 1: |V_n| > u \}$ for the multivariate perpetuity sequence $V_n = Q_1 + M_1 Q_2 + \cdots + (M_1 \ldots M_{n-1}) Q_n$, where $(M_n, Q_n)$ is a sequence of independent and identically distributed random variables with $M_1$ a $d \times d$ ($d \geq 1$) random matrix with nonnegative entries, and $Q_1$ a nonnegative random vector in $\mathbb R^d$. Here $|\cdot|$ denotes the vector norm. The exact asymptotic for the probability $\mathbb P (\tau_u < \infty)$ as $u \to \infty$ has been found by Kesten (Acta Math. 1973). In this paper we prove a conditioned weak law of large numbers for $\tau_u$: conditioned on the event $\{ \tau_u < \infty \}$, $\frac{\tau_u}{\log u}$ converges in probability to a certain constant $\rho > 0$ as $u \to \infty$. A conditioned central limit theorem for $\tau_u$ is also obtained. We further establish precise large deviation asymptotics for the lower probability $\mathbb P (\tau_u \leq (\beta - l) \log u)$ as $u \to \infty$, where $\beta \in (0, \rho)$ and $l \geq 0$ is a vanishing perturbation satisfying $l \to 0$ as $u \to \infty$. Our results extend those of Buraczewski et al. (Ann. Probab. 2016) from the univariate case ($d=1$) to the multivariate case ($d>1$). As consequences, we deduce exact asymptotics for the pointwise probability $\mathbb P (\tau_u = [(\beta - l) \log u] )$ and the local probability $\mathbb P (\tau_u - (\beta - l) \log u \in (a, a + m ] )$, where $a<0$ and $m \in \mathbb Z_+$. We also establish analogous results for the first passage time $\tau_u^y = \inf \{ n \geq 1: \langle y, V_n \rangle > u \}$, where $y$ is a nonnegative vector in $\mathbb R^d$ with $|y| = 1$.