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Asymptotics of the order statistics for a process with a regenerative structure

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  • Soja-Kukieła, Natalia

Abstract

In the paper, a regenerative process {Xn:n∈N} with finite mean cycle length is considered. For Mn(q) denoting the qth largest value in {Xk:1⩽k⩽n}, we prove that supx∈R|PMn(q)⩽x−G(x)n∑k=0q−1−logG(x)nkk!γq,k(x)|→0,asn→∞,for G and γq,k expressed in terms of maxima over the cycle. The result is illustrated with examples.

Suggested Citation

  • Soja-Kukieła, Natalia, 2017. "Asymptotics of the order statistics for a process with a regenerative structure," Statistics & Probability Letters, Elsevier, vol. 131(C), pages 108-115.
    • Handle: RePEc:eee:stapro:v:131:y:2017:i:c:p:108-115
    DOI: 10.1016/j.spl.2017.08.017
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    References listed on IDEAS

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    1. Alpuim, M. Teresa, 1988. "High level exceedances in stationary sequences with extremal index," Stochastic Processes and their Applications, Elsevier, vol. 30(1), pages 1-16, November.
    2. Hsing, Tailen, 1988. "On the extreme order statistics for a stationary sequence," Stochastic Processes and their Applications, Elsevier, vol. 29(1), pages 155-169.
    3. Jakubowski, Adam, 1993. "Asymptotic (r- 1)-dependent representation for rth order statistic from a stationary sequence," Stochastic Processes and their Applications, Elsevier, vol. 46(1), pages 29-46, May.
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