An interesting problem about almost sure convergence Assume that $X_1,X_2,\ldots$ are independent random variables (not necessarily of the same distribution). Assume that that $Var[X_n]>0$ for all $n$. Assume also that
$$\sum_{n=0}^\infty \frac{Var[X_n]}{n^2}<\infty,$$
that
$$\frac{1}{n}\sum_{i=1}^n(X_i-E[X_i])\to 0 \textrm{ almost surely as $n\to\infty$},$$
that
$$E[X_n]>0 \textrm{ for all $n$},$$
and that
$$\liminf_{n\to\infty} E[X_n] > 0.$$
How can we prove that $$\sum_{i=1}^n X_i \to \infty\text{ almost surely as $n\to\infty$?}$$
This seems to intuitively make sense, but a formal proof escapes me. Also, what can we say if $E[X_n] = 0$ for all $n$ and $\lim_{n\to\infty} E[X_n] = 0$?

By the strong law of large numbers, $\frac1n\sum\limits_{k=1}^n\left(X_k-E(X_k)\right)$ converges almost surely. The hypothesis on the expectations implies that $\liminf\limits_{n\to\infty}\frac1n\sum\limits_{k=1}^nE(X_k)\geqslant c$ with $c\gt0$. Summing these two yields $\liminf\limits_{n\to\infty}\frac1n\sum\limits_{k=1}^nX_k\geqslant c$ hence $\sum\limits_{k=1}^nX_k$ diverges to $+\infty$ almost surely (and it does so, at least linearly).
