Here is a version of Doob’s Maximal inequality I want to prove: Fix positive integer $k$. For a real discrete time process $X_n, n=0,1,...,k$, write $X^*:= sup_{n\leq k} X_n$. Now assume $X$ is a real super martingale. Then for $t>0$, $$tP[|X|^*> t]\leq E[|X_0|]+ 2E[|X_k|].$$
Now I’ve proved a different version of Doob's maximal inequality: For X a nonnegative sub martingale, $$tP[X^*> t]\leq E[X_k].$$
Can one derive the former from the latter? Or can someone please point to some other proof of the former
The form of the desired upper bound suggests writing $|X_n |= X_n + 2X_n^-$ and then maybe using the fact that $X_n^-$ is a submartingale since $X_n$ is a supermartingale?
Ok maybe this is it. Let $T:= k \wedge ~min \{n : |X_n|>t\}$. Then T is a stopping time and we have $$ tP[|X|^*> t] \leq E[|X|_T]= E[X_T+2X^-_T]\leq E[X_0]+2E [X^-_T]\leq E[X_0]+2E [X^-_k]\leq E[|X_0|]+2E [|X_k|] $$ where the second inequality is due to $X$ being a supermartingale and the third is due to $X^-$ being a submartingale.