Theorem (Birkhoff) Let $(E, \mathcal{E}, \mu)$ be a $\sigma$-finite measure space and let $T : E \to E$ be a measure-preserving transformation. Suppose that $f \in L^1(\mu)$. Then, the approximations:
$$F_n : =\frac{\sum_{k=0}^{n-1} f \circ T^k}{n}$$
converge almost everywhere to a $T$-invariant integrable function $F:E \to E$
In this theorem, we average the functions $f \circ T^k$ uniformly, but we may note that convergence also holds in some other cases; for instance, if we define:
$$G_n : =\frac{\sum_{k=0}^{n-1} (1+(-1)^k) f \circ T^k}{n}$$
Of course, this particular example can also be proved with Birkhoff by considering $T^2$. However, it raises the question of which coefficients are sufficient for convergence of the ergodic averages.
Question: for each $n$, let $(s^{(n)}_k)_{1 \le k \le n}$ be a vector of non-negative real numbers with $\sum_k s^{(n)}_k = 1$. Suppose that we define:
$$H_n = \sum_k s_k^{(n)} f \circ T^k$$
Which conditions on the family of vectors $(s^{(n)})$ are sufficient for $H_n \to F $ a.e.?