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A general logarithmic asymptotic behavior for partial sums of i.i.d. random variables

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  • Miao, Yu
  • Li, Deli

Abstract

Let 00. Let {X,Xn;n≥1} be a sequence of independent and identically distributed B-valued random variables and set Sn=∑i=1nXi,n≥1. In this note, a general logarithmic asymptotic behavior for {Sn;n≥1} is established. We show that if Sn/n1/p→P0, then, for all s>0, lim supn→∞logP‖Sn‖>sn1/p(logn)θ=−p−θζ¯(p,θ),lim infn→∞logP‖Sn‖>sn1/p(logn)θ=−p−θζ̲(p,θ),where ζ¯(p,θ)=−lim supt→∞logeptP(log‖X‖>t)tθandζ̲(p,θ)=−lim inft→∞logeptP(log‖X‖>t)tθ.The main tools used to prove this result are the symmetrization technique, an auxiliary lemma for the maximum of i.i.d. random variables, a moment inequality, and an exponential inequality.

Suggested Citation

  • Miao, Yu & Li, Deli, 2024. "A general logarithmic asymptotic behavior for partial sums of i.i.d. random variables," Statistics & Probability Letters, Elsevier, vol. 208(C).
    • Handle: RePEc:eee:stapro:v:208:y:2024:i:c:s0167715224000129
    DOI: 10.1016/j.spl.2024.110043
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    References listed on IDEAS

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    1. Gantert, Nina, 2000. "A note on logarithmic tail asymptotics and mixing," Statistics & Probability Letters, Elsevier, vol. 49(2), pages 113-118, August.
    2. Y. Hu & H. Nyrhinen, 2004. "Large Deviations View Points for Heavy-Tailed Random Walks," Journal of Theoretical Probability, Springer, vol. 17(3), pages 761-768, July.
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