Borel-Cantelli lemmas

If $\sum \mathbb{P}(A_n)<\infty$, only finitely many $A_n$ occur almost surely. For independent events, divergence implies infinitely many occur. Use a union bound for the first and $\prod (1-p_n)\le e^{-\sum p_n}$ for the second.

For arbitrary events $A_n$, if ${\sum }_n\mathbb{P}(A_n)<\infty$, then $\mathbb{P}({\mathrm{limsup}}_n{A}_n)=0$. The union bound gives $\mathbb{P}(\mathrm{limsup}{A}_n)\le {\sum }_{n\ge m}\mathbb{P}(A_n)\to 0$. No independence is required.

For a nonnegative random variable $X$, Tonelli theorem givesConsequently, for each $a>0$, $\mathbb{E}[X]=\infty$ exactly when ${\sum }_{n\ge 1}\mathbb{P}(X>an)=\infty$; compare the integral on successive intervals of length $a$.

For independent rate-one exponential variables $X_n$ and $M_n={\max }_{m\le n}{X}_m$,almost surely. The first follows by applying both Borel--Cantelli lemmas to thresholds $(1\pm \epsilon )\log n$; the second uses
$\mathbb{P}(M_n\le (1-\epsilon )\log n)\le e^{-n^{\epsilon }}$.

For events $B_n\in {\mathcal{F}}_n$, put $p_n=\mathbb{P}(B_n\mid {\mathcal{F}}_{n-1})$. Then ${\sum }_n{1}_{B_n}<\infty$ almost surely on $\{{\sum }_n{p}_n<\infty \}$, while ${\sum }_n{1}_{B_n}=\infty$ almost surely on $\{{\sum }_n{p}_n=\infty \}$. The proof applies the Martingale convergence theorem to compensated sums ${\sum }_{j\le n}(1_{B_j}-p_j)$ and uses Kronecker lemma in the divergent case.