The mean convergence almost sure

Question

Let $\{X_n,n\geq 1\}$ be a sequence of random variables such that $X_n\overset{a.s}{\rightarrow}0$.

How to show that: $\frac{X_1+X_2+...+X_n}{n}\overset{a.s}{\rightarrow}0$ ?

I see that, if we can apply the CLT, then the result is right. But they are not iid, so we can not use the CLT, what we need to do? Can we apply the WLLN or SLLN? and How?

Thank in advance.

Answer 1

Actually, this has nothing to do with probability theory; it's basic real analysis.

Lemma If $(x_n)_{n \in \mathbb{N}} \subseteq \mathbb{R}$ is a sequence of real numbers such that $x_n \to 0$, then $$\frac{1}{n} \sum_{j=1}^n x_j \xrightarrow[]{n \to \infty} 0.$$

Proof: For any fixed $\epsilon>0$ we can choose $N \in \mathbb{N}$ such that

$$|x_n| \leq \epsilon \quad \text{for all $n \geq N$.}$$

Thus,

$$\left| \frac{1}{n} \sum_{j=1}^n x_j \right| \leq \frac{1}{n} \sum_{j=1}^N |x_j| + \frac{1}{n} \sum_{j=N+1}^n |x_j| \leq \frac{N}{n} \max_{j \leq N} |x_j| + \epsilon \xrightarrow[]{n \to \infty} \epsilon.$$

As $\epsilon>0$ is arbitrary, this proves the assertion.

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Applying the lemma to $x_n := X_n(\omega)$ (for fixed $\omega$) proves the desired result.