Let us consider a sequence $(S_n)_n$ of $L^2$ random variables.
Assume:
- $S_n \le S_{n+1}$ almost surely
- $S_n \to_{n \to \infty} +\infty$ almost surely
- $\frac{S_n}{\mathbb{E}[S_n]} \to_{n \to \infty} 1$ in the $L^2$ sense.
Is it always true that $\frac{S_n}{\mathbb{E}[S_n]} \to_{n \to \infty} 1$ almost surely?
If not, can you find a counterexample?
PhoemueX's idea does work, here is a way to get an explicit counter-example. Let $X_i$ be defined as $X_i=k_iY_i$, where $(Y_i)_{i\geqslant 1}$ is independent and $Y_i$ takes the value $1$ with probability $1-1/i$ and $2$ with probability $1/i$, and $(k_i)$ is a sequence of integers such that for each $n$, $$ k_n\geqslant n\sum_{i=1}^{n-1}k_i. $$ Let $S_n=\sum_{i=1}^n X_i$. Then $$ 0\leqslant \frac{S_{n-1}}{S_n}\leqslant \frac{2}{k_n}\sum_{i=1}^{n-1}k_i\to 0 $$ and $\mathbb E[S_n]=\sum_{i=1}^nk_i(1+1/i)$ hence $\mathbb E[S_n]\sim \mathbb E[X_n]$. We thus have to check that $\mathbb E\left[\left(X_n/\mathbb E[X_n]-1\right)^2\right]\to 0$ and that $X_n/\mathbb E[X_n]$ does not converge to $1$ almost surely. For the first part, note that $X_n/\mathbb E[X_n]=Y_n/(1+1/n)$ and a computation shows that it goes to $1$ in $L^2$. Moreover, $Y_n/(1+1/n)$ does not converge to $1$ almost surely in view of the second Borel-Cantelli lemma.