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Extremes of q-Ornstein–Uhlenbeck processes

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  • Wang, Yizao

Abstract

Two limit theorems are established on the extremes of a family of stationary Markov processes, known as q-Ornstein–Uhlenbeck processes with q∈(−1,1). Both results are crucially based on the weak convergence of the tangent process at the lower boundary of the domain of the process, a positive self-similar Markov process little investigated so far in the literature. The first result is the asymptotic excursion probability established by the double-sum method, with an explicit formula for the Pickands constant in this context. The second result is a Brown–Resnick-type limit theorem on the minimum process of i.i.d. copies of the q-Ornstein–Uhlenbeck process: with appropriate scalings in both time and magnitude, a new semi-min-stable process arises in the limit.

Suggested Citation

  • Wang, Yizao, 2018. "Extremes of q-Ornstein–Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 2979-3005.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:9:p:2979-3005
    DOI: 10.1016/j.spa.2017.10.008
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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.
    2. Wang, Yizao, 2016. "Large jumps of q-Ornstein–Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 110-116.
    3. Kabluchko, Zakhar, 2009. "Extremes of space-time Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 119(11), pages 3962-3980, November.
    4. 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.
    5. Stoev, Stilian A., 2008. "On the ergodicity and mixing of max-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1679-1705, September.
    6. Cheng, Dan, 2016. "Excursion probability of certain non-centered smooth Gaussian random fields," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 883-905.
    7. Kabluchko, Zakhar & Schlather, Martin, 2010. "Ergodic properties of max-infinitely divisible processes," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 281-295, March.
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    Cited by:

    1. 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).

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