Martingale $X_n \to \infty$ a.s.

Question

Construct a martingale $X_n$ such that $X_n \to \infty $ a.s.

I have trouble coming up with such an example and prove it. Can someone provide an example?

Answer 1

Let $X_1:=0$, $\ X_n:=X_{n-1}-(n-1)+Y_{n}$, where $Y_n$ is independent, $\mathcal{F}_{n}$ measurable and $P(Y_n=0)=1/n$, $P(Y_n=n)=1-1/n$. Then $E[Y_n]=n-1$, so $E[X_n|\mathcal{F}_{n-1}]=X_{n-1}$. Since $X_n\rightarrow\infty$ is a 0-1 event, we just need to show that it occurs with positive probability to conclude $X_n\rightarrow\infty$ almost surely. If $Y_n=n$, then $X_n=X_{n-1}+1$, so clearly

$$P(X_n\rightarrow\infty)\geq \prod_{n=2}^\infty (1-1/n)\geq \lim_{n\rightarrow\infty} (1-1/n)^n=e^{-1}>0$$