Sum of squares of random variables is almost surely finite

Question

Let $\{X_n,n\geq 1\}$ be a sequence of random variables, with the property that $\sum_n \pm_n X_n$ converges a.s. for any choice of $\pm$. I wish to show that $\sum_n X_n^2<\infty$ a.s. (This is exercise 7.7.5 from Resnick's A Probability Path.)

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The idea I have in mind is to find a countable number of sequences $\{b_n^k,n\geq 1\}_{k\in\Bbb N}$ with $b_n^k\in\{-1,1\}$ for all $n,k$, and define $S_k=\{\omega:\sum_n b_n^kX_n(\omega)\text{ converges}\}$. Hopefully for this choice of sequences, $\{\sum_n X_n^2<\infty\}\supseteq \cap_k S_k$, which will prove the result.

Unfortunately I haven't been able to find such a sequence. Any hints or alternative strategies would be appreciated.

Answer 1

The author gives the following hint:

Hint: Consider $B_n(t)X_n(\omega)$ where the random variables $\{B_n\}_{n\geqslant 1}$ are coin tossing or Bernoulli random variables. Apply Fubini on the space of $(t,\omega )$.

Indeed, the main difficulty lies on the fact that we know that for any $\varepsilon= \left(\varepsilon_n\right)\in\{-1,1\}^n$, there exists $\Omega_\varepsilon$ such that $\mathbb P\left(\Omega_\varepsilon\right)=1$ and for all $\omega\in\Omega_\varepsilon$, the series $\sum_{n=1}^{\infty}\varepsilon_n X_n\left(\omega\right)$ converges. But it is not clear whether we can find $\widetilde\Omega$ of probability one which works for any choice of sequences.

Denote $\left(\Omega,\mathcal A,\mathbb P\right)$ the original probability space. On an other probability space $\left(\Omega',\mathcal A',\mathbb P'\right)$, we consider a sequence of independent identically distributed random variables $\left(B_n\right)_{n\geqslant 1}$ such that $\mathbb P'\left(B_n=1\right)=\mathbb P'\left(B_n=-1\right)=1/2$. Denote $$E:=\left\{\left(\omega,\omega'\right): \sum_{n=1}^{+\infty} B_n\left(\omega'\right)X_n\left(\omega \right)\mbox{ does not converge} \right\}.$$ Now appling Fubini's theorem to the indicator function of $E$, we get the existence of $\Omega_0\subset\Omega $ such that $\mathbb P\left(\Omega_0\right)=1$ and for all $\omega\in\Omega_0$, the series $\sum_{n=1}^\infty B_n\left(\omega'\right)X_n\left(\omega\right)$ converges for almost every $\omega\in\Omega'$. Now, use the three series theorem to show that if for some deterministic sequence $\left(a_n\right)_{n\geqslant 1}$, the $\sum_n a_nB_n\left(\omega'\right)$ converges for almost every $\omega\in\Omega'$, then $\sum_n a_n^2$ is finite.