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Suppose that $X_1, X_2, X_3,\ldots$ are sequence of independent random variables such that $\mu_k= 0$ and $ \sigma^2_k =\operatorname{Var}(X_k)< \infty$ for all $k$. Then show that $\sum_{k=1}^{\infty} \sigma^2_k <\infty$ implies $|\sum_{k=1}^{\infty} X_k|<\infty $ almost surely.

I was wondering how could I use the facts: 1) if a martingale $M$ is bounded in $L^2 $ then $\lim M_n$ exists almost surely. 2) Orthogonality of increments of $M$ to prove the above statement. I would like to see the solution in explained way.

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  • $\begingroup$ If $P(|Z| \ge b) \ge a$ then $Var(Z) \ge a b^2$. Using that $Var(Y_n) < C$ and $Var(Y-Y_n) \to 0$ show that the sequence of random variables $Y_n = \sum_{k=1}^n X_k$ is uniformly bounded almost surely, and that it converges almost surely. $\endgroup$
    – reuns
    Apr 7, 2017 at 0:15
  • $\begingroup$ Then where are the orthogonality applied? $\endgroup$
    – 00012 suxn
    Apr 7, 2017 at 11:56
  • $\begingroup$ $Var(Y_n) = \sum_{k=1}^n Var(X_k)$ $\endgroup$
    – reuns
    Apr 7, 2017 at 11:59
  • $\begingroup$ Thank you, Do you have some idea on the following one: math.stackexchange.com/questions/2221753/… $\endgroup$
    – 00012 suxn
    Apr 7, 2017 at 12:11
  • $\begingroup$ Did you complete this question ? Write your own answer then. $\endgroup$
    – reuns
    Apr 7, 2017 at 12:12

1 Answer 1

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Let $M_n:=\sum_{i=1}^nX_i$. Defining $\mathcal F_n$ as the $\sigma$-algebra generated by $X_i$, $1\leqslant i\leqslant n$. Then $(S_n,\mathcal F_n)$ is a martingale and using orthogonality of increments, $$ \mathbb E\left[M_n^2\right]=\sum_{k=1}^n\mathbb E\left[X_k^2\right]=\sum_{k=1}^n\sigma_k^2 $$ hence $$ \sup_n\mathbb E\left[M_n^2\right]\leqslant \sum_{k=1}^{+\infty}\sigma_k^2. $$

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