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Suppose $X_1,X_2,\dots$ are independent random variables such that $EX_n=0$ and $Var(X_n)=\sigma_n^2<\infty$ for all $n\geq 1$. Assume that $\vert X_n(w)\vert \leq K$ for all $n\geq 1$ and all $w\in \Omega$ where $K$ is some non-random positive number. Let $X_0=0, S_0=0, A_0=0$, and $S_n=\sum_{k=1}^n X_k$ and $A_n=\sum_{k=1}^n \sigma_{k}^2$. Suppose $F_n:=\sigma(X_1,\dots,X_n)$ and $F_0=\{\emptyset, \Omega \}$.

Prove the following statement: If $\sum_{n=1}^\infty X_n$ converges almost surely, then $\sum_{n=1}^\infty \sigma_n^2 <\infty$.

Firstly, I showed $S_n^2 -A_n$ is a martingale with respect to $F_n$.

Since $S_n^2 = (S_n^2 -A_n) + (A_n)$ and $S_n^2 - A_n$ is a martingale, I could get $ES_n^2 =EA_n$.

By using the property, I also could get $E(A_{min(n,N)}) \leq (K+c)^2$ for some c>0 where $$N:=\inf\{n\geq 1 : \vert S_n \vert \geq c \}.$$

In Hint, the problem can be solved by the propery $E(A_{\min(n,N)}) \leq (K+c)^2$.

I don't know how to apply the property to my problem...

Any help is appreciated..

Thank you!

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Since

$$\mathbb{E}(A_{\min\{N,n\}}) \leq (K+c)^2$$

it follows from Fatou's lemma that

$$\mathbb{E}(A_{N}) \leq (K+c)^2.$$

As the series $\sum_{n \geq 1} X_n$ converges almost surely, we can choose $c>0$ such that $N=N(c)$ satisfies $\mathbb{P}(N=\infty)>0$. For such $c$ we get

$$\mathbb{P}(N=\infty) \sum_{n \geq 1} \sigma_n^2 \leq \mathbb{E}(A_N) \leq (K+c)^2$$

implying

$$\sum_{n \geq 1} \sigma_n^2 \leq \frac{1}{\mathbb{P}(N=\infty)} (K+c)^2 < \infty.$$

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