Berry–Esseen Bounds with Targets and Local Limit Theorems for Products of Random Matrices

Let $$\mu $$ be a probability measure on $$\textrm{GL}_d(\mathbb {R})$$ 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 . We study statistical properties of variables form $$\begin{aligned} \sigma (S_n,x) + u(S_n x), \end{aligned}$$ $$x \in \mathbb {P}^{d-1}$$ , $$\sigma is norm cocycle u belongs to class admissible functions $$\mathbb values in {R}\cup \{\pm \infty \}$$ Assuming that has finite exponential moment generates proximal strongly irreducible semigroup, we obtain optimal Berry–Esseen bounds Local Limit Theorem for such using large observables {R}$$ Hölder continuous target As particular cases, new limit theorems (S_n,x)$$ coefficients $$S_n$$