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Teoriya Veroyatnostei i ee Primeneniya, 1975, Volume 20, Issue 4, Pages 865–873
(Mi tvp3371)
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This article is cited in 63 scientific papers (total in 63 papers)
Short Communications
The density of the distribution of the maximum of a Gaussian process
B. S. Tsirel'son Leningrad
Abstract:
Let $X(t,\omega)$ be a separable Gaussian process,
\begin{gather*}
\forall t\mathbf EX_t\ge0,\quad f(\omega)=\sup_tX(t,\omega)<\infty\quad\text{with probability }1,
\\
F(a)=\mathbf P\{f<a\};\quad\inf\{a\colon F(a)>0\}=a_0\in[-\infty,+\infty).
\end{gather*}
Then the density $F'(a)$ exists and is continuous at every $a$ except, may be, $a_0$ (at this and only this point $F$ may have a jump!) and at most countable set of points, at which $F'$ has jumps downwards. The density $F'(a)$ decreases almost as rapidly as $\exp(-a^2/2)$ when $a\to+\infty$.
Provided $\mathbf EX_t$ and $\mathbf EX_t^2$ do not depend on $t$, $F$ is continuous everywhere and $F'$ everywhere except, may be, $a_0$, where $F$' may have a jump of a finite size. Asymptotic behaviour of $1-F$ at $+\infty$ determines that of $F'$. Corresponding inequalities are given.
Received: 17.09.1974
Citation:
B. S. Tsirel'son, “The density of the distribution of the maximum of a Gaussian process”, Teor. Veroyatnost. i Primenen., 20:4 (1975), 865–873; Theory Probab. Appl., 20:4 (1976), 847–856
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https://www.mathnet.ru/eng/tvp3371 https://www.mathnet.ru/eng/tvp/v20/i4/p865
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