Estimates of the distribution of the maximum of a random field
Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 2, pp. 350-358
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Let $ \xi(t) $ be a random field with values in $ \mathbb R^1$, defined for $ t \in T$, $T$ an arbitrary set. In this paper two-sided exponential estimates are derived for probabilities $ P(T,u) = \mathbb P\{\sup_{t \in T} \xi(t) > 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 $u$.
Keywords:
entropy, spaces $ B(\varphi)$, entropy germcapacity, exponential estimate.
@article{TVP_1997_42_2_a8,
author = {E. I. Ostrovskii},
title = {Estimates of the distribution of the maximum of a~random field},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {350--358},
year = {1997},
volume = {42},
number = {2},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1997_42_2_a8/}
}
E. I. Ostrovskii. Estimates of the distribution of the maximum of a random field. Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 2, pp. 350-358. http://geodesic.mathdoc.fr/item/TVP_1997_42_2_a8/