Local limit theorem for the operator norm of products of random matrices

We prove a local limit theorem for the quantity $\mathrm{\parallel }{g}_n\cdots g_1\mathrm{\parallel }$, where $g_1,g_2,\dots$ is a sequence of independent and identically distributed random invertible square matrices. The norm $\mathrm{\parallel }\cdot \mathrm{\parallel }$ is arbitrary and we make no proximality assumption.