Suppose $\eta_t$ is a pathwise Riemann integrable stochastic process, with $t\in[0,1]$. Consider, for every partition $\{t_{j,n}=\frac{j}{n}|~j=0,...,n\}$ of $[0,1]$, the Riemann sum
$$ S_n=\sum_{j=0}^n\eta_{t_{j,n}}\,\varepsilon_{j,n}\frac{1}{n} $$
where $\varepsilon_{j,n}$ is a triangular array of random variables. In the case the $\varepsilon_{j,n}$'s are iid with expected value $\mu$ it is immediate to prove that
$$ S_n\stackrel{p}{\to}\mu\,\int_0^1\eta_s\,ds $$
Now assume that the $\varepsilon_{j,n}$ are not iid. Is it possible somehow to apply the Birkhoff Ergodic Theorem?