Sum of centered random variables with finite variance converges almost surely

Question

Given $X_i$ for $i \geq 1$ random variables with mean 0 and $\sum_{i=1}^{\infty}\operatorname{Var}(X_i)<\infty$
and their sum $S_n = \sum_{i=1}^{n}X_i$ I want to show that $S_n$ converges almost surely as $n \to \infty$ using the maximum difference between any two elements of $S_n$ is finite, using: $P\left(\max_{1 \leq k \leq r} |S_{n+k} – S_n| > \epsilon\right)$ Any hints?

Answer 1

I am going to prove that $(S_n)$ is almost surely a Cauchy sequence -- which will imply that $(S_n)$ converges almost surely, as your random variables are real-valued.

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Consider $Z_n=\sup_{k,l\ge n} |S_k-S_l|$. We want to show that $Z_n\to 0$ almost surely. From Kolmogorov's inequality,

$$\mathsf{P}(Z_n\ge \varepsilon)\le \frac{2}{\varepsilon^2} \sum_{k\ge n} \mathsf{E} X_k^2.$$

Since $\sum_k \mathsf{E} X_k^2$ converges by assumption, for all $r\ge 1$ one can find $n_r\ge 1$ such that $\mathsf{P}(Z_{n_r}\ge 1/r)\le 1/r^2$. By Borel-Cantelli lemma, almost surely it holds that $Z_{n_r}<1/r$ for all but finitely many $r$. Finally $(Z_n)$ is nonincreasing, so that $Z_n\to 0$ almost surely.