Conditioned local limit theorems for products of positive random matrices

Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identically distributed positive random $d\times d$ matrices, where $d\geq 2$ 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 $\tau_{x, y} = \inf \{ k \geq 1: y + \log |g_k \cdots g_1 x| < 0 \}$. 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 $n^{-3/2}$.