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Limit laws for the maxima of stationary chi-processes under random index

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  • Zhongquan Tan
  • Changchun Wu

Abstract

Let $$\{\chi _{k}(t), t\ge 0\}$$ { χ k ( t ) , t ≥ 0 } be a stationary $$\chi $$ χ -process with $$k$$ k degrees of freedom. In this paper, we consider the maxima $$M_{k}(T)= \max \{\chi _{k}(t), \forall t\in [0,T]\}$$ M k ( T ) = max { χ k ( t ) , ∀ t ∈ [ 0 , T ] } with random index $$\mathcal {T}_{T}$$ T T , where $$\mathcal {T}_{T}/T$$ T T / T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of $$M_{k}(\mathcal {T}_{T})$$ M k ( T T ) exists under some additional conditions. Copyright Sociedad de Estadística e Investigación Operativa 2014

Suggested Citation

  • Zhongquan Tan & Changchun Wu, 2014. "Limit laws for the maxima of stationary chi-processes under random index," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(4), pages 769-786, December.
    • Handle: RePEc:spr:testjl:v:23:y:2014:i:4:p:769-786
    DOI: 10.1007/s11749-014-0366-6
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    References listed on IDEAS

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    1. Stamatovic, Biljana & Stamatovic, Sinisa, 2010. "Cox limit theorem for large excursions of a norm of a Gaussian vector process," Statistics & Probability Letters, Elsevier, vol. 80(19-20), pages 1479-1485, October.
    2. A. Freitas & J. Hüsler & M. Temido, 2012. "Limit laws for maxima of a stationary random sequence with random sample size," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(1), pages 116-131, March.
    3. 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.

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