Conditioned local limit theorems for products of positive random matrices
Abstract
Consider the random matrix products $G_n: = g_n \ldots g_1$, where $(g_{n})_{n\geq 1}$ is a sequence of independent and identically distributed positive random $d\times d$ matrices for any integer $d \geq 2$. For any starting point $x \in \mathbb R_+^d$ with $|x| = 1$ and $y \geq 0$, consider the exit time $\tau_{x, y} = \inf \{ k \geq 1: y + \log |G_k x| < 0 \}$. In this paper, we study the conditioned local probability $\mathbb P ( y + \log |G_n x| \in [0, \Delta] + z, \, \tau_{x, y} > n )$ under various assumptions on $y$ and $z$. For $y = o(\sqrt{n})$, we prove precise upper and lower bounds when $z$ is in a compact interval and give exact asymptotics when $z \to \infty$. We also study the case when $y \asymp \sqrt{n}$ and establish the corresponding asymptotics in function of the behaviour of $z$.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2023
- DOI:
- arXiv:
- arXiv:2310.07565
- Bibcode:
- 2023arXiv231007565G
- Keywords:
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- Mathematics - Probability
- E-Print:
- 52 pages. arXiv admin note: text overlap with arXiv:2110.05123