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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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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/

[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

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