Consider the matrix products \(G_n: = g_n \cdots g_1\) , where \((g_{n})_{n\geqslant 1}\) is a sequence of independent and identically distributed positive random \(d\times d\) matrices. Under the optimal third moment condition, we first establish a Berry–Esseen theorem and an Edgeworth expansion for the (i, j)-th entry \(G_n^{i,j}\) of the matrix \(G_n\) , where \(1 \leqslant i, j \leqslant d\) . Utilizing the Edgeworth expansion for \(G_n^{i,j}\) under the changed probability measure, we then prove precise upper and lower large deviation asymptotics for the entries \(G_n^{i,j}\) subject to an exponential moment assumption. As applications, we deduce local limit theorems with large deviations for \(G_n^{i,j}\) and establish upper and lower large deviations bounds for the spectral radius \(\rho (G_n)\) of \(G_n\) . A byproduct of our approach is the local limit theorem for \(G_n^{i,j}\) under the optimal second moment condition. In the proofs we develop a spectral gap theory for both the norm cocycle and the coefficients, which is of independent interest.
