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

Question

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.

Answer 1

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.