Limit theorems for first passage times of multivariate perpetuity sequences

TL;DR

This paper investigates the asymptotic behavior of the first passage times for multivariate perpetuity sequences, extending known univariate results to higher dimensions and establishing limit theorems and large deviation principles.

Contribution

It extends the asymptotic analysis of first passage times from univariate to multivariate perpetuity sequences, providing new limit theorems and precise large deviation results.

Findings

Conditioned weak law of large numbers for u/u converges to  > 0

Conditioned central limit theorem for u

Precise large deviation asymptotics for u

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 $|y| = 1$.