{
  "id": "2307.04985",
  "version": "v1",
  "published": "2023-07-11T02:51:30.000Z",
  "updated": "2023-07-11T02:51:30.000Z",
  "title": "Limit theorems for first passage times of multivariate perpetuity sequences",
  "authors": [
    "Sebastian Mentemeier",
    "Hui Xiao"
  ],
  "categories": [
    "math.PR"
  ],
  "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 $\\mathbb R^d$. 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 $u \\to \\infty$. Even in the case $d = 1$, this improves the previous large deviation result by Buraczewski et al. (Ann. Probab. 2016). 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 $|y| = 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-11T02:51:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first passage time",
      "multivariate perpetuity sequence",
      "limit theorem",
      "probability",
      "exact asymptotic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}