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Extremes of the standardized Gaussian noise

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  • Kabluchko, Zakhar

Abstract

Let be a d-dimensional array of independent standard Gaussian random variables. For a finite set define . Let A be the number of elements in A. We prove that the appropriately normalized maximum of , where A ranges over all discrete cubes or rectangles contained in {1,...,n}d, converges in law to the Gumbel extreme-value distribution as n-->[infinity]. We also prove a continuous-time counterpart of this result.

Suggested Citation

  • Kabluchko, Zakhar, 2011. "Extremes of the standardized Gaussian noise," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 515-533, March.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:3:p:515-533
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    Cited by:

    1. Guenther Walther & Andrew Perry, 2022. "Calibrating the scan statistic: Finite sample performance versus asymptotics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(5), pages 1608-1639, November.
    2. Kabluchko, Zakhar & Wang, Yizao, 2014. "Limiting distribution for the maximal standardized increment of a random walk," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2824-2867.
    3. Zhongquan Tan & Shengchao Zheng, 2020. "Extremes of a Type of Locally Stationary Gaussian Random Fields with Applications to Shepp Statistics," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2258-2279, December.
    4. Hotz, Thomas & Marnitz, Philipp & Stichtenoth, Rahel & Davies, Laurie & Kabluchko, Zakhar & Munk, Axel, 2012. "Locally adaptive image denoising by a statistical multiresolution criterion," Computational Statistics & Data Analysis, Elsevier, vol. 56(3), pages 543-558.

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