Short Communications: Estimates of the Distribution of the Maximumof a Random Field

Let $ \xi(t) $ be a random field with values in $ \bR^1$, defined for $ t \in T,\ T$ an arbitrary set. In this paper two-sided exponential estimates are derived for probabilities $ P(T,u) = \bP\{\sup_{t \in T} \xi(t)\break > u \} $: $$ C_1 g_2(u) \l \log P(T,\,u) + g_1(u) \l C_2 g_2(u), $$ where...

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Veröffentlicht in:Theory of probability and its applications 1998-04, Vol.42 (2), p.302-310
1. Verfasser: Ostrovskii, E. I.
Format: Artikel
Sprache:eng
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Zusammenfassung:Let $ \xi(t) $ be a random field with values in $ \bR^1$, defined for $ t \in T,\ T$ an arbitrary set. In this paper two-sided exponential estimates are derived for probabilities $ P(T,u) = \bP\{\sup_{t \in T} \xi(t)\break > u \} $: $$ C_1 g_2(u) \l \log P(T,\,u) + g_1(u) \l C_2 g_2(u), $$ where $ g_1(u) $ is a convex function, $u \to \infty \Rightarrow \lim g_1'(u) = \infty$, $\lim [g_2(u)/g_1(u)] = 0$, $C_k$ are positive numbers independent of~$u$.
ISSN:0040-585X
1095-7219
DOI:10.1137/S0040585X97976167