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Feb
27
comment Martingale and bounded stopping time
@ByronSchmuland Yes.
Feb
27
comment Martingale and bounded stopping time
@ByronSchmuland Probability: Theory and Examples, 4th edition
Feb
27
awarded  Promoter
Feb
26
comment Martingale and bounded stopping time
I gave my method another try and it didn't work. So I decided to offer a bounty.
Feb
25
revised Martingale and bounded stopping time
added 73 characters in body
Feb
24
asked Martingale and bounded stopping time
Feb
24
accepted Generalized Second Borel-Cantelli lemma
Feb
23
asked Generalized Second Borel-Cantelli lemma
Feb
22
awarded  Critic
Feb
21
revised Proving existence of limit by Martingale.
added 660 characters in body
Feb
21
asked Convergence of Martingale.
Feb
21
comment Proving existence of limit by Martingale.
@Ilya You're right. But this question appears as an exercise for martingales. So there must be some way that we can relate this fixed sequence to a super or sub martingale. I'm thinking that could there be $X_n$ such that $X_n$ converges almost surely, and $\prod_m (1+y_m)$ is one of $X_n$'s realization.
Feb
21
answered A fair coin flip, which of the events are independent
Feb
21
comment Proving existence of limit by Martingale.
@ByronSchmuland Yeah, you are right. There is a problem if $M > 1$.
Feb
21
asked Proving existence of limit by Martingale.
Feb
20
accepted Martingale that converges almost surely to $-\infty$.
Feb
20
comment Martingale that converges to zero
@ColinMcQuillan I see! Because $Y_1,Y_2,..$ are i.i.d., so we can find an $\epsilon > 0$ such that $\mathbb P[|Y_1 - 1| ~> \epsilon] < 1$ for all $n$. Therefore $\mathbb P[Y_n = 1] = 0$. So here $i.i.d.$ is a necessary condition.
Feb
20
comment Martingale that converges to zero
@ColinMcQuillan But couldn't there exists a set $A = \{\omega:Y_n(\omega) \to 1\}$ such that $\mathbb P\{A\} > 0$?
Feb
20
accepted Martingale that converges to zero
Feb
20
asked Martingale that converges to zero