Asymptotic properties of the norm of the extremum of a sequence of normal random functions
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 }\li...
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| Datum: | 1998 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
1998
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/4822 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510998613458944 |
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| author | Matsak, I. K. Мацак, І. К. |
| author_facet | Matsak, I. K. Мацак, І. К. |
| author_sort | Matsak, I. K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:15:12Z |
| description | 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. |
| first_indexed | 2026-03-24T03:05:54Z |
| format | Article |
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| id | umjimathkievua-article-4822 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:05:54Z |
| publishDate | 1998 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fa/4d2493fcc5b7f01d58b32fe792472dfa.pdf |
| spelling | umjimathkievua-article-48222020-03-18T21:15:12Z Asymptotic properties of the norm of the extremum of a sequence of normal random functions Асимптотичні властивості норми екстремуму послідовності нормальних випадкових функцій Matsak, I. K. Мацак, І. К. 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. При додаткових умовах па обмежену нормально розподілену випадкову функцію $X = X( t),\; t ∈ T$ встановлено співвідношення вигляду $$\mathop {\lim }\limits_{n \to \infty } P(b_n (||Z_n || - a_n ) \leqslant x) = \exp ( - e^{ - x} )\forall x \in R^1$$ де $Z_n = Z_n (t) = \mathop {\max }\limits_{1 \leqslant k \leqslant n} X_k (t), \;(X_n )$-незалежні копії $X,||x(t)|| = \mathop {\sup }\limits_{1 \in T} |x(t)|$ — $(a_n), (b_n)$ числові послідовності. Institute of Mathematics, NAS of Ukraine 1998-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4822 Ukrains’kyi Matematychnyi Zhurnal; Vol. 50 No. 10 (1998); 1359–1365 Український математичний журнал; Том 50 № 10 (1998); 1359–1365 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4822/6343 https://umj.imath.kiev.ua/index.php/umj/article/view/4822/6344 Copyright (c) 1998 Matsak I. K. |
| spellingShingle | Matsak, I. K. Мацак, І. К. Asymptotic properties of the norm of the extremum of a sequence of normal random functions |
| title | Asymptotic properties of the norm of the extremum of a sequence of normal random functions |
| title_alt | Асимптотичні властивості норми екстремуму послідовності нормальних
випадкових функцій |
| title_full | Asymptotic properties of the norm of the extremum of a sequence of normal random functions |
| title_fullStr | Asymptotic properties of the norm of the extremum of a sequence of normal random functions |
| title_full_unstemmed | Asymptotic properties of the norm of the extremum of a sequence of normal random functions |
| title_short | Asymptotic properties of the norm of the extremum of a sequence of normal random functions |
| title_sort | asymptotic properties of the norm of the extremum of a sequence of normal random functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4822 |
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