TITLE: Convergence of series of random variables with random signs.
QUESTION [7 upvotes]: Suppose that $\{X_{n}, n\ge 1\}$ are arbitrary random variables such that $\sum_{n}\pm X_{n}$ converges almost surely for any choice of signs $\pm 1$.  Show that $\sum X_{n}^2<\infty$ almost surely. [Hint: Consider $\sum_{n} B_{n}(t)X_{n}(\omega)$ where the random variables $\{B_{n},n\ge 1\}$ are Bernoulli.  Apply Fubini on the space of $(t,\omega)$.]
Remarks 

This is problem 7.7.4(b) of Resnick's A Probability Path.
We know that if $\sum \pm a_{i}$ converges almost surely, then $\sum a_{i}^2$ converges  if the $a_{i}$ are fixed (non-random).  This is discussed  here, for example. It's also Lemma 7.6.1 of Resnick.
The point of the problem seems to be to pass from knowledge of the result for fixed values of the $X_{n}$ to random values; and presumably that's the point of the hint about Fubini.  But I don't understand exactly what's needed to draw this conclusion.

REPLY [2 votes]: Let $A = \{(t,w):\sum B_n(t) X_n(w)\}$ converges.  For each $w$, let $A_w = \{t : (t,w) \in A\}$, and for each $t$, let $A^t = \{w:(t,w) \in A\}$.
By hypothesis, for almost every $t$, the set $A^t$ has measure 1.  By Fubini, 
$$ \int \text{measure}(A_w) \, dw = \int \text{measure}(A^t) \, dt ,$$
since both sides equal $\int \!\!\int \chi_A(t,w) \, dt \, dw$.
Hence for almost every $w$, the set $A_w$ has measure 1.  But by your point 2, for any $w$, the set $A_w$ has measure 1 only if $\sum |X_n(w)|^2 < \infty$.
Thus for almost every $w$, we have $\sum |X_n(w)|^2 < \infty$.