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We say that $X_1,X_2,.... $ is a completely convergent sequence to $X$ if

$$ \sum_{n=1}^{\infty} P( |X_n -X | > \epsilon ) < \infty \; \; \; \; for \; \; each \; \; \epsilon >0$$

Question: If sequence of r.v ($X_n)$ is independent, can we conclude that the notions of completely convergence and convergence a.s are equivalent ?

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If $(X_n)_{n \in \mathbb{N}}$ is completely convergent, then it follows from the (first) Borel-Cantelli lemma that $X_n \to X$ almost surely.

On the other hand, if $(X_n)_{n \in \mathbb{N}}$ is a sequence of independent random variables such that $X_n \to X$ almost surely, then $X=c$ almost surely for some constant $c \in \mathbb{R}$. Therefore, the events

$$\{|X_n-X|>\epsilon\}, \qquad n \in \mathbb{N},$$

are independent. It follows from the (second) Borel-Cantelli lemma that $X_n$ converges completely to $X$.

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