Edgeworth Expansion and Large Deviations for the Coefficients of Products of Positive Random Matrices

Consider the matrix products $$G_n: = g_n \cdots g_1$$ G n : = g n ⋯ g 1 , where $$(g_{n})_{n\geqslant 1}$$ ( g n ) n ⩾ 1 is a sequence of independent and identically distributed positive random $$d\times d$$ d × 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 $$G_n^{i,j}$$ G n i , j of the matrix $$G_n$$ G n , where $$1 \leqslant i, j \leqslant d$$ 1 ⩽ i , j ⩽ d . Utilizing the Edgeworth expansion for $$G_n^{i,j}$$ G n i , j under the changed probability measure, we then prove precise upper and lower large deviation asymptotics for the entries $$G_n^{i,j}$$ G n i , j subject to an exponential moment assumption. As applications, we deduce local limit theorems with large deviations for $$G_n^{i,j}$$ G n i , j and establish upper and lower large deviations bounds for the spectral radius $$\rho (G_n)$$ ρ ( G n ) of $$G_n$$ G n . A byproduct of our approach is the local limit theorem for $$G_n^{i,j}$$ 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.