Sur les Conditions de l'emploi de la loi forte des Grands Nombres des Processus Stationaires de Second Ordre

I. N. Verbickaya

Teoriâ veroâtnostej i ee primeneniâ, Tome 9 (1964) no. 2, pp. 358-365

Résumé

Let $\xi(t)$ be a continuous second order stationary process and ${\mathbf M}\xi(t)=0$, ${\mathbf M}\xi(t)\xi(s)=R(t-s)=R(\tau)$. In order for $$ \lim_{T\to\infty}\frac1T\int_0^T\xi(t)\,dt=0 $$ to hold with probability one, it suffices that $$ \int_1^\infty\frac{\lg^2t}t|\bar R(t)|\,dt\infty, $$ where $$ |\bar R(t)|=\frac1t\int_0^t R(\tau)\,d\tau. $$ If for almost all $t$, $|\xi(t)|\leqq k$, then in order that $$ \lim_{T\to\infty}\frac1T\int_0^T\xi(t)\,dt=0 $$ with probability one, is suffices that $$ \int_1^\infty\frac{|\bar R(t)|}t\,dt\infty. $$