I'm trying to solve the following exercise:
Let $(X_n)_n $ be a collection of i.i.d $L^2$ random variables, with $E[X_n]=0 ,E[X_n^2]=1$ and let $S_n = X_1 + \ldots X_n$.
Show that can't exists a random variable $X$ such that $S_n \rightarrow X$ a.s.
Hint: If $S_n$ converges, then $X_n \rightarrow 0$ a.s.
First question:
- I can't undestand the hint: Why if $S_n$ converges, then $X_n \rightarrow 0$ a.s. ?
- Second question is about my attempt, just using the hint
Attempt:
Since $\sup_{n \in \mathbb{N} } E[X_n^2] = 1$, I have that the martingale $(X_n)_n$ is $L^2$-bounded, therefore converges a.s. and in $L^2$. By hint it converges to $0$, and the $L^2$ convergence implies that $\lim_{n} E[X_n^2] = E[0]$.
But that's a contradiction since the limit on the left is exactly $1$, while the rhs is $0$.
Is it okay?
Here is an alternative proof which does not use the hint. Suppose for contradiction that $S_n$ converged a.s. to some random variable $X$. Then $X/\sqrt{n}$ converges a.s. to $0$. But this contradicts the central limit theorem, which states that $S_n/\sqrt{n}$ converges a.s. to an $N(0,1)$ random variable.