Edgeworth expansion for the coefficients of random walks on the general linear group

Let $(g_n{)}_{n\ge 1}$ be a sequence of independent and identically distributed random elements with law $\mu$ on the general linear group $\text{GL}(V)$, where $V={\mathbb{R}}^d$. Consider the random walk $G_n:=g_n\dots g_1$, $n\ge 1$. Under suitable conditions on $\mu$, we establish the first-order Edgeworth expansion for the coefficients $⟨f,G_{n}v⟩$ with $v\in V$ and $f\in V^{\ast }$, in which a new additional term appears compared to the case of vector norm $\mathrm{\parallel }{G}_{n}v\mathrm{\parallel }$.