The density of the distribution of the maximum of a~Gaussian process

B. S. Tsirel'son

Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 4, pp. 865-873

Résumé

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\};\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.