On the maximum of a Gaussian stationary process

M. G. Šur

Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 2, pp. 386-389

Résumé

We consider a Gaussian real process $x(t)$ which satisfies the same conditions as in [1]. We prove the existence (a.s.) of such random number $t_0$ ($t_0<\infty$) that the inequality $$ |\max_{o\le u\le t}x(u)-\sigma\sqrt{2\ln t}|<\frac{(\sigma+\varepsilon)\ln\ln t}{\sqrt{2\ln t}} $$ is valid for all $t>t_0$ where $\varepsilon$ is any fixed positive number and $\sigma^2=\mathbf Mx^2(t)$.