For which processes do we have $$ \lim_{T\to\infty}\frac{1}{T}\int_{0}^{T} X_t dt =\lim_{a\to 0}a\int_{0}^{\infty} e^{-at}X_t dt $$ almost surely?
The Hardy Littlewood implies that this holds for processes that are bounded below or above. But I feel like there is a non trivial generalization under suitable autocorrelation assumptions.