[Math] Borel-Cantelli Lemma and almost sure convergence

Question

Almost sure convergence is usually proved by Borel-Cantelli lemma. If the condition of Borel-Cantelli lemma doesn't hold, does the almost sure convergence still hold? If so, how to construct a sequence of identically distributed random variables $\{X_n,n\geq1\}$ and a sequence of real numbers $0\leq a_n\rightarrow\infty$ such that $$\sum_{n=1}P[|X_n|>a_n]=\infty,\mbox{ and } \ X_n/a_n\rightarrow 0\mbox{ a.s.} $$

Answer 1

Just consider $X_n=X$ where $X$ is not integrable. Then for any sequence $\left(a_n\right)_{n\geqslant 1}$ such that $a_n\to +\infty$, $X_n/a_n=X/a_n$ goes to $0$ as $n$ goes to infinity. Now, for the choice $a_n=n$ or something which growth slower, we have $\sum_{n=1}^{+\infty}\Pr\left\{X_n> a_n\right\}= +\infty$.