Conditioned local limit theorems for products of positive random matrices

Abstract

Let $(g_n{)}_{n\ge 1}$ be a sequence of independent and identically distributed positive random $d\times d$ matrices, where $d\ge 2$ is an integer. For any starting point $x\in {\mathbb{R}}_{+}^d$ with $|x|=1$ and $y\in \mathbb{R}$, we define the exit time ${\tau }_{x,y}=\inf \{k\ge 1:y+\log |g_k\cdots g_{1}x|<0\}$. In this paper, we investigate the conditioned local probability $\mathbb{P}(y+\log |g_n\cdots g_{1}x|\in z+[0,\Delta ],{\tau }_{x,y}>n)$ under various assumptions on $y$, $z$ and $\Delta$. For the case where $z=O(\sqrt{n})$, we establish an exact asymptotic result as $n\to \infty$, uniformly in $y$ and $\Delta$, which extends the classical Caravenna conditioned local limit theorem to the case of products of positive random matrices. Our proof does not rely on the reversibility techniques. Furthermore, for arbitrary $z\in {\mathbb{R}}_{+}$, we deduce a uniform upper bound with rate $n^{-3/2}$.

arXiv comment33 pages. arXiv admin note: text overlap with arXiv:2110.05123