2
$\begingroup$

I have the assumption that $\{X_n\} \geq 0$ is a supermartingale such that for some real $\epsilon >0$, either $X_{n+1}=X_n$ or $|X_{n+1} - X_n| \geq \epsilon$ a.s. for all $n\in\mathbb{Z_+}$.

When $T := \sup\{n: |X_{n+1} - X_n| \geq \epsilon\} \in \mathbb{Z_+}\cup\{\infty\}$ is the time of last trial, how should I prove that $P(T < \infty)=1$ and $E[X_T]\leq E[X_0]$? I believe that I need to use the optional stopping theorem, but still feel very perplexed about what I need to consider and show first.

Thanks for everyone's help!

  • 3
    Optional stopping theorem doesn't apply, because $T$ isn't a stopping time with respect to $(X_n)_{n\in\mathbb{N}}$. To be a stopping time, the event $\{T\leq n\}$ must be measurable with respect to $\sigma(X_1,\ldots,X_n)$. – 2017-01-27
  • 0
    @NickPeterson Thank you. I was confused with the definition of stopping time. Then may I ask you how to approach this question instead? – 2017-01-27

0 Answers 0