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Exact asymptotics and limit theorems for supremum of stationary χ-processes over a random interval

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  • Tan, Zhongquan
  • Hashorva, Enkelejd

Abstract

Let {χk(t),t≥0} be a stationary χ-process with k degrees of freedom being independent of some non-negative random variable T. In this paper we derive the exact asymptotics of P{supt∈[0,T]χk(t)>u} as u→∞ when T has a regularly varying tail with index λ∈[0,1). Three other novel results of this contribution are the mixed Gumbel limit law of the normalised maximum over an increasing random interval, the Piterbarg inequality and the Seleznjev pth-mean theorem for stationary χ-processes.

Suggested Citation

  • Tan, Zhongquan & Hashorva, Enkelejd, 2013. "Exact asymptotics and limit theorems for supremum of stationary χ-processes over a random interval," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 2983-2998.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:8:p:2983-2998
    DOI: 10.1016/j.spa.2013.03.009
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    References listed on IDEAS

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    1. Arendarczyk, Marek & Dȩbicki, Krzysztof, 2012. "Exact asymptotics of supremum of a stationary Gaussian process over a random interval," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 645-652.
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    6. Piterbarg, V. I., 1994. "High excursions for nonstationary generalized chi-square processes," Stochastic Processes and their Applications, Elsevier, vol. 53(2), pages 307-337, October.
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    Cited by:

    1. Krzysztof Dȩbicki & Enkelejd Hashorva & Lanpeng Ji & Chengxiu Ling, 2015. "Extremes of order statistics of stationary processes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 229-248, June.
    2. Qiao, Wanli, 2021. "Extremes of locally stationary Gaussian and chi fields on manifolds," Stochastic Processes and their Applications, Elsevier, vol. 133(C), pages 166-192.
    3. Das, Bikramjit & Engelke, Sebastian & Hashorva, Enkelejd, 2015. "Extremal behavior of squared Bessel processes attracted by the Brown–Resnick process," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 780-796.

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