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I want to know weather a non negative supermartingale converges to $0$.

I have a hunch that it shall be so, but could not prove or disprove it. Is this correct? And if so, is there a way to prove it?

PacMafs
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athos
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    Easy counterexample: Let $X_0$ be a non-negative $\mathcal F_0$-measurable random variable that isn't identically $0$. The sequence $(X_0,X_0,...)$ is a non-negative supermartingale that doesn't converge to $0$. – user6247850 Oct 12 '21 at 14:34
  • @KaviRamaMurthy I'm referring to super-martingale, not submartingale. – athos Oct 12 '21 at 20:43
  • @user6247850 how about a strict supermartingale? – athos Oct 12 '21 at 20:44

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No, this does not hold. If $(X_n)$ is a non-negative supermartingale converging to $0$, then $Y_n := X_n+1$ is still a supermartingale and $\lim (Y_n) = \lim(X_n)+1 = 1$.

user6247850
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