I have two questions, somehow related. The first one, has to do with the Birkhoff's ergodic theorem. In its classical formulation, it states that if we have a probability space $(X,\mathcal{B},\mu)$ and a measure preserving transformation $T:X\to X$, then for every $f\in L^1$ the averages $$ \dfrac{1}{n}\sum_{k=0}^{n-1}f\circ T^k (x) $$ converge to a function $\hat{f}\in L^1$, invariant by $T$. While the classical proofs (for example, the ones found in Walters, Ward-Einsiedler, etc) make use of the maximal inequality and the maximal ergodic theorem, it is not clear to me why the normalization by $n$ is the right one. Is it possible to normalize by other sequences $a_n\nearrow\infty$? In the sense that $$ (*)\qquad \dfrac{1}{a_n}\sum_{k=0}^{n-1}f\circ T^k (x) $$ could provide some additional information, or maybe not.

My second question is essentially the same, but in the infinite measure context. Jon Aaronson proved in *On the ergodic theory of non-integrable functions and infinite measure spaces*, 1977 that the averages $(*)$ do not provide any useful information regardless of the choice of the sequence $a_n$. While it is possible to follow the calculations, I do not get the idea behind this result. Any insight about this would be very appreciated. Thanks in advance.