Moderate deviations and local limit theorems for the coefficients of random walks on the general linear group

Consider the random walk $G_n:=g_n\dots g_1$, $n\ge 1$, where $(g_n{)}_{n\ge 1}$ is a sequence of independent and identically distributed random elements with law $\mu$ on the general linear group $\mathrm{G}\mathrm{L}(V)$ with $V={\mathbb{R}}^d$. Under suitable conditions on $\mu$, we establish Cramér type moderate deviation expansions and local limit theorems with moderate deviations for the coefficients $⟨f,G_{n}v⟩$, where $v\in V$ and $f\in V^{\ast }$. Our approach is based on the Hölder regularity of the invariant measure of the Markov chain $G_n\text{\hspace{0.17em}⁣}\cdot \text{\hspace{0.17em}⁣}x=\mathbb{R}{G}_{n}v$ on the projective space of $V$ with the starting point $x=\mathbb{R}v$, under the changed measure.