A map $f: X \to X$ preserves an ergodic probability $\mu$, i.e., $\mu \circ f^{-1}=\mu$ and for any $\phi: X \to \mathbb{R}$ with $\int \phi d\mu=0$, $$\frac{1}{n} \sum_{i \le n} \phi \circ f^i \to 0 \text{ almost surely and in } L^1(\mu).$$

Therefore, $\max_{n \ge N} \frac{1}{n} \sum_{i \le n} \phi \circ f^i \to 0$ almost surely.

Similar to maximal inequality, are there references to quantitatively study

$$\max_{n \ge N} \frac{1}{n} \sum_{i \le n} \phi \circ f^i \text{ ?}$$

Similar to martingale inequality, are there references to study

$$\lVert\max_{n \ge N} \frac{1}{n} \sum_{i \le n} \phi \circ f^i\rVert_{L^1} \precsim \lVert\frac{1}{N} \sum_{i \le N} \phi \circ f^i\rVert_{L^1} \text{ ?}$$