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A second moment bound for critical points of planar Gaussian fields in shrinking height windows

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  • Muirhead, Stephen

Abstract

We consider the number of critical points of a stationary planar Gaussian field, restricted to a large domain, whose heights lie in a certain interval. Asymptotics for the mean of this quantity are simple to establish via the Kac–Rice formula, and recently Estrade and Fournier proved a second moment bound that is optimal in the case that the height interval does not depend on the size of the domain. We establish an improved bound in the more delicate case of height windows that are shrinking with the size of the domain.

Suggested Citation

  • Muirhead, Stephen, 2020. "A second moment bound for critical points of planar Gaussian fields in shrinking height windows," Statistics & Probability Letters, Elsevier, vol. 160(C).
    • Handle: RePEc:eee:stapro:v:160:y:2020:i:c:s0167715220300018
    DOI: 10.1016/j.spl.2020.108698
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    References listed on IDEAS

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    1. Estrade, Anne & Fournier, Julie, 2016. "Number of critical points of a Gaussian random field: Condition for a finite variance," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 94-99.
    2. Cammarota, V. & Wigman, I., 2017. "Fluctuations of the total number of critical points of random spherical harmonics," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 3825-3869.
    3. Nicolaescu, Liviu I., 2017. "A CLT concerning critical points of random functions on a Euclidean space," Stochastic Processes and their Applications, Elsevier, vol. 127(10), pages 3412-3446.
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