Geodesic
New result on the behaviour of Gaussian maxima in terms of the covariance function
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 32, Tome 510 (2022), pp. 201-210 
S. M. Novikov. New result on the behaviour of Gaussian maxima in terms of the covariance function. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 32, Tome 510 (2022), pp. 201-210. http://geodesic.mathdoc.fr/item/ZNSL_2022_510_a11/
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     title = {New result on the behaviour of {Gaussian} maxima in terms of the covariance function},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2022_510_a11/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

It is a well-known result by Berman [1] that if the covariance function $r(n)$ of a stationary centered Gaussian sequence tends to zero as $n$ tends to infinity, then the maximum of its first $n$ elements is $\sqrt{2\log(n)}(1+o(1))$ almost surely. In this work we discuss whether or not the Cesàro convergence of $|r(n)|$ to zero necessarily implies the same asymptotic.

[1] S. M. Berman, “Limit theorems for the maximum term in stationary sequences”, Ann. Math. Statist., 35 (1964), 502–516 | DOI | MR

[2] I. P. Cornfeld, Ya. G. Sinai, S. V. Fomin, Ergodic Theory, Nauka, M., 1980 | MR

[3] J. L. Doob, Stochastic Processes, Wiley, New York, 1953 | MR

[4] R. J. Adler, J. E. Taylor, Random Fields and Geometry, Springer, New York, 2007 | MR

[5] A. Tadevosyan, Gaussian Assignment Process, bachelor thesis, 2021

[6] G. Mordant and J. Segers, “Maxima and near-maxima of a Gaussian random assignment field”, Statist. Probab. Lett., 173 (2021), 109087, 8 pp. | DOI | MR