Assume that $W$ is a zero-mean Gaussian process on $[0,\infty)$ with covariance function satisfying $c(s,t):=E(W(s)W(T))=C(|t-s|)$ for a function $C:[0,\infty) \to \mathbb{R}$.
Can I exploit somehow the Ergodic theorem to claim that (in probability or almost surely) $$ \frac{1}{\log(K_n/k_n)}\sum_{j=k_n}^{K_n} \frac{1}{j} W^2(\log(j/K_n)) \to C(0) $$ for two integer sequences $k_n \leq K_n$ such that $K_n/k_n \to \infty$ as $n \to \infty$? I'm used to apply the Erdogic theorem to claim that, for a stationary sequence $X_1, X_2, \ldots$ with $E|X_1|<\infty$, almost surely as $T \to \infty$, $$ \frac{1}{T}\sum_{t=1}^T X_t \to E(X_1), $$ see e.g. Theorem 6.28 in Breiman's Probability book of 1992. Can I relate the latter convergence to the former?