0
$\begingroup$

Does anybody know a reference for the following theorem?

Theorem 1. Let $(X_t)_{t=0}^\infty$ be a non-negative supermartingale.
Then, for any constant $c > 0$, the event $(\exists > t)\, X_t \ge c$ has probability at most $E[X_0]/c$.

The theorem generalizes the standard Markov bound.

The theorem is not hard to prove, but I haven't been able to find it in the literature.

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ This post: math.stackexchange.com/questions/4868201/… refers to your inequality as Doob's Maximal inequality for a positive supermartingale. $\endgroup$
    – Anthony Quas
    Commented Nov 30 at 17:21
  • $\begingroup$ Thank you! From that name I was able to find a suitable reference here: en.wikipedia.org/wiki/… : Revuz & Yor 1999, Corollary II.1.6 and Theorem II.1.7. If you'd like to make an answer out of this please go ahead. If not I can do it. $\endgroup$
    – Neal Young
    Commented Nov 30 at 19:22
  • 1
    $\begingroup$ Please go ahead. $\endgroup$
    – Anthony Quas
    Commented Nov 30 at 19:24
  • $\begingroup$ Correction. The citation I gave is incorrect. The correct citation is to the same text, but Page 58, "(1.15) Exercise (Maximal inequality for positive supermartingales)." Also, I apparently am unable to post an answer (probably I don't have enough reputation). $\endgroup$
    – Neal Young
    Commented Nov 30 at 19:34

0

Reset to default

You must log in to answer this question.