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Self-normalized Cramér type moderate deviations for stationary sequences and applications

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Listed:
  • Fan, Xiequan
  • Grama, Ion
  • Liu, Quansheng
  • Shao, Qi-Man

Abstract

Let (Xi)i≥1 be a stationary sequence. Denote m=⌊nα⌋,0<α<1, and k=⌊n∕m⌋, where ⌊a⌋ stands for the integer part of a. Set Sj∘=∑i=1mXm(j−1)+i,1≤j≤k, and (Vk∘)2=∑j=1k(Sj∘)2. We prove a Cramér type moderate deviation expansion for P(∑j=1kSj∘∕Vk∘≥x) as n→∞. Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed.

Suggested Citation

  • Fan, Xiequan & Grama, Ion & Liu, Quansheng & Shao, Qi-Man, 2020. "Self-normalized Cramér type moderate deviations for stationary sequences and applications," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 5124-5148.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:8:p:5124-5148
    DOI: 10.1016/j.spa.2020.03.001
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    References listed on IDEAS

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    1. Grama, Ion & Haeusler, Erich, 2000. "Large deviations for martingales via Cramér's method," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 279-293, February.
    2. Gao, Fu-Qing, 1996. "Moderate deviations for martingales and mixing random processes," Stochastic Processes and their Applications, Elsevier, vol. 61(2), pages 263-275, February.
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    Cited by:

    1. Cui, Jiazhen & Liu, Qiaojing, 2023. "Cramér-type moderate deviations for the log-likelihood ratio of inhomogeneous Ornstein–Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 192(C).
    2. Doukhan, Paul & Fan, Xiequan & Gao, Zhi-Qiang, 2023. "Cramér moderate deviations for a supercritical Galton–Watson process," Statistics & Probability Letters, Elsevier, vol. 192(C).

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