Let $X_1, X_2, \cdots$ be a sequence of $p$-integrable $\mathbb{R^d}$ valued random variables. Assume that $X_n$ converges $0$ weakly, then can we say that $r(\omega) = liminf\{ |X_1 (\omega)|, |X_2 (\omega)|, \cdots\}$ is $0$ for almost every $\omega ?$
The classical example of a weakly convergent but not strongly convergent sequence is those of orthogonal basis, yet it is not a counterexample for the above claim. I could not prove the claim but intuitively I believe it holds, at least for $\mathbb{R^d}$ valued random vectors.
If needed one can assume that the sequence $(X_n)_n$ is bounded in $p$-norm, I do not feel like this is necessary.
Weak convergence together with boundedness of second moments implies convergence of expectations. By Fatou's Lemma we get $E \lim \inf |X_n| \leq \lim \inf E|X_n|=0$ since $E|X_n| \to 0$. This implies that $ \lim \inf |X_n|=0$ almost surely.