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Asymptotic properties of the norm of the extremum of a sequence of normal random functions

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Abstract

Under additional conditions on a bounded normally distributed random function X = X( t), t ∈ T, we establish a relation of the form

$$\mathop {\lim }\limits_{n \to \infty } P(b_n (||Z_n || - a_n ) \leqslant x) = \exp ( - e^{ - x} )\forall x \in R^1 $$

where \(Z_n = Z_n (t) = \mathop {\max }\limits_{1 \leqslant k \leqslant n} X_k (t),(X_n )\) are independent copies of \(X,||x(t)|| = \mathop {\sup }\limits_{1 \in T} |x(t)|\), and (a n) and (b n) are numerical sequences.

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References

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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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Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 10, pp. 1359–1365, October, 1998.

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Matsak, I.K. Asymptotic properties of the norm of the extremum of a sequence of normal random functions. Ukr Math J 50, 1551–1558 (1998). https://doi.org/10.1007/BF02513503

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  • DOI: https://doi.org/10.1007/BF02513503

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