A conditioned local limit theorem for non-negative random matrices

TL;DR

This paper establishes a conditioned local limit theorem for non-negative random matrices, providing asymptotic estimates for the probability that the associated process stays positive and within a specific set up to a certain time.

Contribution

It introduces a new conditioned local limit theorem for non-negative matrix products, extending previous results with asymptotic bounds for the process's behavior.

Findings

Derived asymptotic estimates for the process staying positive and in a compact set.

Provided bounds on the probability of the process's trajectory up to time n.

Extended the strategy of Denisov and Wachtel to matrix product processes.

Abstract

Let $(S_n)_n$ be the random process on $\mathbb R$ driven by the product of i.i.d. non-negative random matrices and $\tau$ its exit time from $]0, +\infty[$. By using the adapted strategy initiated by D. Denisov and V. Wachtel, we obtain an asymptotic estimate and bounds of the probability that the process $(S_k)_k$ remains non negative up to time $n$ and simultaneously belongs to some compact set $[b, b+\ell ]\subset \mathbb R^{*+}$ at time $n$.