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Extremal behavior of squared Bessel processes attracted by the Brown–Resnick process

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  • Das, Bikramjit
  • Engelke, Sebastian
  • Hashorva, Enkelejd

Abstract

The convergence of properly time-scaled and normalized maxima of independent standard Brownian motions to the Brown–Resnick process is well-known in the literature. In this paper, we study the extremal functional behavior of non-Gaussian processes, namely squared Bessel processes and scalar products of Brownian motions. It is shown that maxima of independent samples of those processes converge weakly on the space of continuous functions to the Brown–Resnick process.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:2:p:780-796
    DOI: 10.1016/j.spa.2014.09.006
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    References listed on IDEAS

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    1. Engelke, S. & Kabluchko, Z. & Schlather, M., 2011. "An equivalent representation of the Brown-Resnick process," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1150-1154, August.
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    6. 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.
    7. Hüsler, Jürg & Liu, Regina Y. & Singh, Kesar, 2002. "A Formula for the Tail Probability of a Multivariate Normal Distribution and Its Applications," Journal of Multivariate Analysis, Elsevier, vol. 82(2), pages 422-430, August.
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    Cited by:

    1. Engelke, Sebastian & Kabluchko, Zakhar, 2015. "Max-stable processes and stationary systems of Lévy particles," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4272-4299.
    2. 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).
    3. E. Hashorva, 2018. "Approximation of Some Multivariate Risk Measures for Gaussian Risks," Papers 1803.06922, arXiv.org, revised Oct 2018.

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