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Moderate deviations and local limit theorems for the coefficients of random walks on the general linear group

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  • Xiao, Hui
  • Grama, Ion
  • Liu, Quansheng

Abstract

Consider the random walk Gn:=gn…g1, n⩾1, where (gn)n⩾1 is a sequence of independent and identically distributed random elements with law μ on the general linear group GL(V) with V=Rd. Under suitable conditions on μ, we establish Cramér type moderate deviation expansions and local limit theorems with moderate deviations for the coefficients 〈f,Gnv〉, where v∈V and f∈V∗. Our approach is based on the Hölder regularity of the invariant measure of the Markov chain Gn⋅x=RGnv on the projective space of V with the starting point x=Rv, under the changed measure.

Suggested Citation

  • Xiao, Hui & Grama, Ion & Liu, Quansheng, 2023. "Moderate deviations and local limit theorems for the coefficients of random walks on the general linear group," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 103-133.
    • Handle: RePEc:eee:spapps:v:158:y:2023:i:c:p:103-133
    DOI: 10.1016/j.spa.2022.12.013
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    References listed on IDEAS

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    1. Xiao, Hui & Grama, Ion & Liu, Quansheng, 2020. "Precise large deviation asymptotics for products of random matrices," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5213-5242.
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