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

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
$$ \int_1^\infty\frac{|\bar R(t)|}t\,dt<\infty. $$