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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
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