2019
Том 71
№ 9

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

Matsak I. K.

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.

English version (Springer): Ukrainian Mathematical Journal 50 (1998), no. 10, pp 1551-1558.

Citation Example: Matsak I. K. Asymptotic properties of the norm of the extremum of a sequence of normal random functions // Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1359–1365.

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