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Asymptotic properties of the norm of extremum values of normal random elements in the space C[0, 1]

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Ukrainian Mathematical Journal Aims and scope

Abstract

We prove that

$$\mathop {\lim }\limits_{n \to \infty } \left( {\left\| {Z_n } \right\| - (2 ln (n))^{1/2} \left\| \sigma \right\|} \right) = 0 a.s.,$$

where X is a normal random element in the space C [0,1], MX = 0, σ = {(M¦X(t)¦2)1/2 t∈[0,1}, (X n ) are independent copies of X, and \(Z_n = \mathop {\max }\limits_{l \leqslant k \leqslant n} X_k \). Under additional restrictions on the random element X, this equality can be strengthened.

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Authors and Affiliations

  1. Ukrainian State Academy of Light Industry, Kiev

    I. K. Matsak

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  1. I. K. Matsak
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1227–1235, September, 1998.

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Matsak, I.K. Asymptotic properties of the norm of extremum values of normal random elements in the space C[0, 1]. Ukr Math J 50, 1405–1415 (1998). https://doi.org/10.1007/BF02525246

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  • DOI: https://doi.org/10.1007/BF02525246

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