If $\sum_{n}\sqrt{\mathbb{E}(X_{n+1}-X_n)^2}<\infty$, show that r.v. sequence $\{X_n\}$ converges almost surely and in $L^1$.

Question

Consider a r.v. sequence $\{X_n\}$. Prove that if $\sum_{n}\sqrt{\mathbb{E}(X_{n+1}-X_n)^2}<\infty$ then $X_n$ converges almost surely and in $L^1$ norm.

I know that $\sum_{n}\sqrt{\mathbb{E}(X_{n+1}-X_n)^2}<\infty$ implies $\sum_{n}\mathbb{E}|X_{n+1}-X_n|<\infty$, then $\mathbb{E}|X_{n+1}-X_n|\to 0$, but it is failed to get $X_n$ converges. So I think $L^2$ norm in this series is necessary and sufficient to achieve the proof, but how?

Answer 1

Recall that

Proposition. If $(\mathcal{X}, \|\cdot\|)$ is a Banach space, and if $(x_n)$ is a sequence in $\mathcal{X}$ such that $$ \sum_{n=1}^{\infty} \|x_{n+1} - x_n\| < \infty, $$ then $(x_n)$ converges in $\mathcal{X}$.

Now using the inequality $\mathbf{E}[|Z|]\leq\mathbf{E}[Z^2]^{1/2}$ for any integrable r.v. $Z$, we get

$$\sum_{n=1}^{\infty} \mathbf{E}[|X_{n+1}-X_n|] \leq \sum_{n=1}^{\infty} \mathbf{E}[(X_{n+1}-X_n)^2]^{1/2}<\infty.$$

By the proposition applied to $\mathcal{X} = L^1(\mathbf{P})$, we conclude the $L^1$-convergence. Moreover, either MCT or Tonelli's theorem gives

$$ \mathbf{E}\left[ \sum_{n=1}^{\infty} |X_{n+1}-X_n| \right] = \sum_{n=1}^{\infty} \mathbf{E}[|X_{n+1}-X_n|] < \infty, $$

hence $\sum_{n=1}^{\infty} |X_{n+1}-X_n| < \infty$ holds a.s. Then the a.s.-convergence follows from the proposition applied to $\mathcal{X} = \mathbb{R}$.