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

• Published in: Journal of theoretical probability Vol. 38; no. 2; p. 38
• Main Authors: Xiao, Hui, Grama, Ion, Liu, Quansheng
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2025; Springer Nature B.V
• Subjects: Central limit theorem; Deviation; Euclidean space; Hypotheses; Mathematics; Mathematics and Statistics; Probability Theory and Stochastic Processes; Random variables; Statistics; Theorems; Berry–Esseen theorem; Primary 60F05; Spectral gap; 60F10; Edgeworth expansion; Precise large deviations; Secondary 60J05; 60B20; Products of positive random matrices; Local limit theorem
• ISSN: 0894-9840; 1572-9230
• DOI: 10.1007/s10959-025-01406-z

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.