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On the probability of conjunctions of stationary Gaussian processes

Author

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  • Dȩbicki, Krzysztof
  • Hashorva, Enkelejd
  • Ji, Lanpeng
  • Tabiś, Kamil

Abstract

Let {Xi(t),t≥0},1≤i≤n be independent centered stationary Gaussian processes with unit variance and almost surely continuous sample paths. For given positive constants u,T, define the set of conjunctions C[0,T],u≔{t∈[0,T]:min1≤i≤nXi(t)≥u}. Motivated by some applications in brain mapping and digital communication systems, we obtain exact asymptotic expansion of P{C[0,T],u≠ϕ}, as u→∞. Moreover, we establish the Berman sojourn limit theorem for the random process {min1≤i≤nXi(t),t≥0} and derive the tail asymptotics of the supremum of each order statistics process.

Suggested Citation

  • Dȩbicki, Krzysztof & Hashorva, Enkelejd & Ji, Lanpeng & Tabiś, Kamil, 2014. "On the probability of conjunctions of stationary Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 141-148.
    • Handle: RePEc:eee:stapro:v:88:y:2014:i:c:p:141-148
    DOI: 10.1016/j.spl.2014.02.004
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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.
    2. Debicki, Krzysztof, 2002. "Ruin probability for Gaussian integrated processes," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 151-174, March.
    3. Worsley, K. J. & Friston, K. J., 2000. "A test for a conjunction," Statistics & Probability Letters, Elsevier, vol. 47(2), pages 135-140, April.
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    Cited by:

    1. Nadarajah, Saralees, 2015. "Complete asymptotic expansions for normal extremes," Statistics & Probability Letters, Elsevier, vol. 103(C), pages 127-133.
    2. Dȩbicki, Krzysztof & Hashorva, Enkelejd & Ji, Lanpeng & Tabiś, Kamil, 2015. "Extremes of vector-valued Gaussian processes: Exact asymptotics," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4039-4065.
    3. Tang, Linjun & Zheng, Shengchao & Tan, Zhongquan, 2021. "Limit theorem on the pointwise maxima of minimum of vector-valued Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 176(C).
    4. K. Dębicki & K. M. Kosiński, 2018. "An Erdös–Révész Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process," Journal of Theoretical Probability, Springer, vol. 31(1), pages 579-597, March.

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