Let be a sequence of independent and identically distributed (i.i.d.) real random matrices. For set . Given any starting point , consider the Markov chain on the projective space and define the norm cocycle by , for an arbitrary norm on . Under suitable conditions we prove a Berry–Esseen-type theorem and an Edgeworth expansion for the couple . These results are established using a brand new smoothing inequality on complex plane, the saddle point method and additional spectral gap properties of the transfer operator related to the Markov chain . Cramér-type moderate deviation expansions as well as a local limit theorem with moderate deviations are proved for the couple with a target function on the Markov chain .
Cite this article
Hui Xiao, Ion Grama, Quansheng Liu, Berry–Esseen bound and precise moderate deviations for products of random matrices. J. Eur. Math. Soc. 24 (2022), no. 8, pp. 2691–2750DOI 10.4171/JEMS/1142