Conditioned local limit theorems for products of positive random matrices

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 $\mathrm{\mid }x\mathrm{\mid }=1$ and $y\in \mathbb{R}$, we define the exit time ${\tau }_{x,y}=\inf \{k\ge 1:y+\log \mathrm{\mid }{g}_k\cdots g_{1}x\mathrm{\mid }<0\}$. In this paper, we investigate the conditioned local probability $\mathbb{P}(y+\log \mathrm{\mid }{g}_n\cdots g_{1}x\mathrm{\mid }\in z+[0,\mathrm{\Delta }],{\tau }_{x,y}>n)$ under various assumptions on $y$, $z$ and $\mathrm{\Delta }$. For the case where $z=O(\sqrt{n})$, we establish an exact asymptotic result as $n\to \mathrm{\infty }$, uniformly in $y$ and $\mathrm{\Delta }$, 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 $n^{-3\mathrm{/}2}$.