Berry–Esseen bound and precise moderate deviations for products of random matrices

Let ${(g{n}){n\\geq 1}}$ be a sequence of independent and identically distributed (i.i.d.) ${d\\times d}$ real random matrices. For ${n\\geq 1}$ set ${G_n = g_n \\ldots g_1}$. Given any starting point ${x=\\mathbb R v\\in\\mathbb{P}^{d-1}}$, consider the Markov chain ${X_n^x \\mathbb G_n v }$ on projective space ${\\mathbb P^{d-1}}$ define norm cocycle by ${\\sigma(G_n, x)= \\log (|G_n v|/|v|)}$, for an arbitrary ${|\\cdot|}$ $\\smash{\\mathbb R^{d}}$. Under suitable conditions we prove Berry–Esseen-type theorem Edgeworth expansion couple ${(X_n^x, \\sigma(G_n, x))}$. These results are established using brand new smoothing inequality complex plane, saddle method additional spectral gap properties transfer operator related to ${X_n^x}$. Cramér-type moderate deviation expansions as well local limit with deviations proved x))}$ target function ${\\varphi}$