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Let $p \in [0 , \frac{1}{2}] $ and $\eta_{i}$ be i.i.d random variables and $P(\eta_{i}=1)=p$ and $P(\eta_{i}=-1)=1-p$ and $\mathcal F_{n}=\sigma(\eta_{1},\cdots,\eta_{n})$ and $X_{n}=\sum_{i=1}^{n}\eta_{i}$ and $Y_{n}=X_{T(-7) \wedge n} $ . show that

1) $(X_{n})_{n\in \mathbb N}$ and $(Y_{n})_{n\in \mathbb N}$ are supermartingale or matigale or submartingale.

2) $(Y_{n})_{n\in \mathbb N}$ is almost surely convergence to $-7$.

thanks for any help.

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Wait, you cannot prove that X is a supermartingale? –  Did Nov 25 '13 at 17:23
    
@Did.i edit the question I dont know which one is true. –  pual ambagher Nov 25 '13 at 17:32
    
@Did.can you help me to solve this problem? –  pual ambagher Nov 25 '13 at 17:41
    
$E[X_{n+1}|\mathcal{F}_n]=E[\eta_{n+1}|\mathcal{F}_n]+E[X_n|\mathcal{F}_n]$. What can you figure out about each term? I bet that helps you with the first part of 1. Now what does T mean to you? –  Henrik Nov 25 '13 at 18:35
    
Thanks @Henrik. –  Did Nov 25 '13 at 20:55
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1 Answer

up vote 4 down vote accepted

OP has solved question 1 himself, see the comments. Regarding question 2:

Let $T(-7)=\inf\{n\geq 1 : X_n\in (-\infty,-7] \}$. Let us first make a quick argument that $P(T(-7)<\infty)=1$ which maybe you know.

By the law of large numbers $X_n/n \to p-(1-p) < 0$ a.s. if $p\in[0,1/2)$ so $X_n \to - \infty$ a.s. giving $P(T(-7)<\infty)=1$. For $p=1/2$ it is a (i hope) well known fact that $P(X_n \text{ hits m before -7})=-7/(-7-m) \to 0$ and by monotone convergence $P(T(-7)<\infty)=1-\lim_{m} P(X_n \text{ hits m before -7})=1$.

Then $\lim_{n} Y_{n} = \lim _{n} X_{T(-7)\wedge n} =X_{T(-7)}=-7$ a.s per properties of hitting times.

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