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Berry–Esseen bounds and moderate deviations for random walks on GLd(R)

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

Abstract

Let (gn)n⩾1 be a sequence of independent and identically distributed random elements of the general linear group GLd(R), with law μ. Consider the random walk Gn:=gn…g1. Denote respectively by ‖Gn‖ and ρ(Gn) the operator norm and the spectral radius of Gn. For log‖Gn‖ and logρ(Gn), we prove moderate deviation principles under exponential moment and strong irreducibility conditions on μ; we also establish moderate deviation expansions in the normal range [0,o(n1/6)] and Berry–Esseen bounds under the additional proximality condition on μ. Similar results are found for the couples (Xnx,log‖Gn‖) and (Xnx,logρ(Gn)) with target functions, where Xnx:=Gn⋅x is a Markov chain and x is a starting point on the projective space P(Rd).

Suggested Citation

  • Xiao, Hui & Grama, Ion & Liu, Quansheng, 2021. "Berry–Esseen bounds and moderate deviations for random walks on GLd(R)," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 293-318.
  • Handle: RePEc:eee:spapps:v:142:y:2021:i:c:p:293-318
    DOI: 10.1016/j.spa.2021.08.005
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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.
    2. Huang, Chunmao & Liu, Quansheng, 2012. "Moments, moderate and large deviations for a branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 522-545.
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