Equivalence of Convergence of Sum of Random Variables

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Let $\{X_n\}$ be a sequence of independent random variables, then $$\sum_nX_n\mbox{ converges a.e.}\iff\sum_nX_n\mbox{ converges in probability}.$$

$\bullet$ Proof.

Since the almost sure convergence implies convergence in probability, we only need to prove the $(\Longleftarrow)$ part.  Let $$S_{m,n}=\sum_{j=m+1}^nX_j.$$Given $0<\varepsilon<1$, the convergence of $S_{m,n}$ in probability can be written as $$\exists\,m_0\mbox{ such that for }m_0<k<n,\,\mathscr{P}\{|S_{k,n}|>\varepsilon\}<\varepsilon.$$For $m<k\leq n$, we have $$\bigcup_{k=m+1}^n\left\{\underset{m<j\leq k-1}{\max}|S_{m,j}|\leq2\varepsilon;\,|S_{m,k}|>2\varepsilon;\,|S_{k,n}|\leq\varepsilon\right\}\subset\left\{|S_{m,n}|>\varepsilon\right\}.$$Since the sets in the union are disjoint and the independence of $S_{k,n}$ to the others, the probability $$\sum_{k=m+1}^n\mathscr{P}\left\{\underset{m<j\leq k-1}{\max}|S_{m,j}|\leq2\varepsilon;\,|S_{m,k}|>2\varepsilon\right\}\mathscr{P}\left\{|S_{k,n}|\leq\varepsilon\right\}\leq\mathscr{P}\left\{|S_{m,n}|>\varepsilon\right\}.$$Then $$\begin{array}{rl}\mathscr{P}\left\{|S_{m,n}|>\varepsilon\right\}&\geq\sum_{k=m+1}^n\mathscr{P}\left\{\underset{m<j\leq k-1}{\max}|S_{m,j}|\leq2\varepsilon;\,|S_{m,k}|>2\varepsilon\right\}\mathscr{P}\left\{|S_{k,n}|\leq\varepsilon\right\}\\
&\geq(1-\varepsilon)\sum_{k=m+1}^n\mathscr{P}\left\{\underset{m<j\leq k-1}{\max}|S_{m,j}|\leq2\varepsilon;\,|S_{m,k}|>2\varepsilon\right\}\\
&\qquad\left(\because\,\mathscr{P}\left\{|S_{k,n}|\leq\varepsilon\right\}\geq1-\varepsilon\right)\\
&\geq(1-\varepsilon)\mathscr{P}\left\{\bigcup_{k=m+1}^n\left(\underset{m<j\leq k-1}{\max}|S_{m,j}|\leq2\varepsilon;\,|S_{m,k}|>2\varepsilon\right)\right\}\\
&\geq(1-\varepsilon)\mathscr{P}\left\{\underset{m<j\leq n}{\max}|S_{m,j}|>2\varepsilon\right\}.\end{array}$$Thus, by the monotone property of sets, $$\begin{array}{rl}\mathscr{P}\left\{|S_n|>2\varepsilon\mbox{ i.o.}\right\}&=\underset{n\rightarrow\infty}{\lim}\mathscr{P}\left\{\bigcap_{m}\bigcup_{j=m+1}^n|S_{m,j}|>2\varepsilon\right\}\\
&\leq\underset{n\rightarrow\infty}{\lim}\mathscr{P}\left\{\bigcup_{j=m+1}^n|S_{m,j}|>2\varepsilon\right\}\\
&\leq\underset{n\rightarrow\infty}{\lim}\frac{\mathscr{P}\{|S_{m,n}|>\varepsilon\}}{1-\varepsilon}\\
&\leq\frac{\varepsilon}{1-\varepsilon}\equiv\varepsilon',\mbox{ for all }n\mbox{ and }\varepsilon. \end{array}$$Thus, $\sum_nX_n$ converges a.e.

$\Box$