Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $\tau:\Omega\to\Omega$ be a measurable map on $(\Omega,\mathcal A)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$, $Y_n:\Omega\to[-\infty,\infty)$ be $\mathcal E$-measurable for $n\in\mathbb N$ with $\operatorname E\left[Y_1^+\right]<\infty$ and $$Y_{m+n}\le Y_m+Y_n\circ\tau^m\;\;\;\text{for all }m,n\in\mathbb N\tag1$$ and $$M_n:=\max(Y_1,\ldots,Y_n)\;\;\;\text{for }n\in\mathbb N.$$

It's easy to show the following extension of the maximal ergodic theorem: $$\operatorname E[Y_1;M_n\ge0]\ge0\;\;\;\text{for all }n\in\mathbb N.\tag2$$

The ordinary maximal ergodic theorem is given by the special case, where $$Y_n=\sum_{i=0}^{n-1}X\circ\tau^i\;\;\;\text{for all }n\in\mathbb N$$ for some integrable real-valued random variable on $(\Omega,\mathcal A,\operatorname P)$. In that special case, it can be deduced from $(2)$ that $$\operatorname P\left[\sup_{n\in\mathbb N}\left|\frac{Y_n}n\right|\ge c\right]\le\frac1c\operatorname E[|Y_1|]\;\;\;\text{for all }c>0\tag3.$$

Can we extend this result to the general case?