The ``Kolmogorov's sub-martingale inequality" says the following : that if $\{ Z_n \mid n = 1,..\}$ is a non-negative sub-martingale then for any $a >0$ and $m \in \mathbb{Z}^+ $we have, $\mathbb{P} [ \max_{n=1,..m} Z_n > a ] \leq \frac{\mathbb{E} [Z_m ]}{a}$

• I would like to know if there are more refined versions of this.

• Like is there a version of this for bounded difference super-martingales with the ``max" replaced by "min"?

• Is there a version of this for bounded difference super-martingales with the ``max" replaced by say the average, $(1/m) \sum_{n=1}^m Z_n$ ?