Berry-Esseen bounds with targets and Local Limit Theorems for products of random matrices

Let $\mu$ be a probability measure on ${\text{GL}}_d(\mathbb{R})$ and denote by $S_n:=g_n\cdots g_1$ the associated random matrix product, where $g_j$’s are i.i.d.'s with law $\mu$. We study statistical properties of random variables of the form $$\sigma (S_n,x)+u(S_{n}x),$$ where $x\in {\mathbb{P}}^{d-1}$, $\sigma$ is the norm cocycle and $u$ belongs to a class of admissible functions on ${\mathbb{P}}^{d-1}$ with values in $\mathbb{R}\cup \{\pm \mathrm{\infty }\}$. Assuming that $\mu$ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we obtain optimal Berry-Esseen bounds and the Local Limit Theorem for such variables using a large class of observables on $\mathbb{R}$ and H"older continuous target functions on ${\mathbb{P}}^{d-1}$. As particular cases, we obtain new limit theorems for $\sigma (S_n,x)$ and for the coefficients of $S_n$.