Edgeworth Expansion and Large Deviations for the Coefficients of Products of Positive Random Matrices - Xiao, Hui; Grama, Ion; Liu, Quansheng (Journal of theoretical probability, 2025)
Consider the matrix products G n : = g n ⋯ g 1 , where ( g n ) n ⩾ 1 is a sequence of independent and identically distributed positive random 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 of the matrix G n , where 1 ⩽ i , j ⩽ 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 ρ ( 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.