# How to prove $(X_{n})_{n\in \mathbb N}$ and $(Y_{n})_{n\in \mathbb N}$ are supermartingale and $(Y_{n})_{n\in \mathbb N}$ is convergence to -7

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 $Y_{n}=X_{T(-7) \wedge n}$(means martingale/submartingale/supermartingale when stopped by a stopping time) . show that

1. $(X_{n})_{n\in \mathbb N}$ and $(Y_{n})_{n\in \mathbb N}$ are supermartingale or matigale or submartingale.
• @Alecos Papadopoulos.subscript in $Y_{n\in \mathbb N}$ means stopping time. – pual ambagher Nov 26 '13 at 5:23
• @Alecos Papadopoulos.I edit the question – pual ambagher Nov 26 '13 at 7:25
• Is this homework or self-study? If so, please add the appropriate tag. – jbowman Nov 26 '13 at 15:15

$$E(X_{n+1}\mid \mathcal F_{n}) = E(\eta_{n+1}+X_{n}\mid \mathcal F_{n})$$ $$=E(\eta_{n+1}\mid \mathcal F_{n}) +E(X_{n}\mid \mathcal F_{n}) =E(\eta_{n+1}) + X_n$$
Analogously for $Y_n$ - if you understand its subscript (I don't).
• .subscript in $Y_{n\in \mathbb N}$ means stopping time. – pual ambagher Nov 26 '13 at 5:22
• can you solve the $Y_{n}$ – pual ambagher Jan 3 '14 at 9:04
• You have deleted the part referring to $Y_n$, and also, you have received an answer for $Y_n$ in math.SE, an answer that you have accepted. Why do you need another answer? – Alecos Papadopoulos Jan 3 '14 at 11:29