I wonder if there are any $L_p$ maximal inequalities for supermartingales. Specifically, for a non-negative supermartingale $\{X_n\}$ and $p\geq 1$, is there any $A_p, B_p>0$ such that $$\mathbb{E}\left[\sup_{0\leq k\leq n} |X_k|^p\right] \leq A_p \mathbb{E}\left[ |X_0|^p\right] + B_p \mathbb{E}[|X_n|^p] .$$

Or is there any inequalities so that we can bound the LHS?

  • $\begingroup$ Yes, there is a similar inequality for supermartingales. The proof is almost the same as the one for submartingales, so you might want to try looking at the proof of the inequality for submartingales and seeing if you can generalize it. $\endgroup$ Jun 24, 2021 at 17:56
  • $\begingroup$ @user6247850 It will be extremely helpful if you can point me to any reference of it. The only trouble I have is the convex function of a supermartingale is no longer a supermartingale. $\endgroup$
    – Kenneth Ng
    Jun 24, 2021 at 18:30
  • $\begingroup$ Sorry, I may have made a mistake - I was thinking of only $L^1$ $\endgroup$ Jun 24, 2021 at 18:54
  • $\begingroup$ $L^1$ is also fine. I don't really need all $p\geq 1$. $\endgroup$
    – Kenneth Ng
    Jun 24, 2021 at 19:26


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