When does $\sum_{i=1}^{\infty} X_i$ exist for random sequences $\{X_i\}_{i=1}^{\infty}$? Suppose $\{X_1, X_2, X_3, \ldots\}$ is an infinite sequence of random variables such that $E[X_i]=0$ for all $i$, and $E[X_iX_j]=0$ whenever $i \neq j$.  Further suppose the variances $\sigma_i^2 = E[X_i^2]$ are finite and satisfy $\sum_{i=1}^{\infty} \sigma_i^2 < \infty$.  
Define $S_n = \sum_{i=1}^nX_i$.  When can one conclude that $S_n$ converges almost surely as $n\rightarrow\infty$? 
I can show that, with prob 1, the limit exists (and is a real number) over a particular subsequence $n[m]$, so that $\lim_{m\rightarrow\infty}S_{n[m]}$ exists as a (random) real number.  I suspect that, in general, we do not have convergence.  However, counter-examples seem hard.
 A: This answer is first adapted from comments in: does convergence in $L^p$ imply convergence almost everywhere.  Consider a probability space $[0, 1]$ with uniform measure.  Define the random variable $X_{2^i + k}$ by
$$X_{2^i + k} = 2^{i/2}\chi_{\left[\frac{k}{2^i}, \frac{k+1/2}{2^i}\right]} - 2^{i/2}\chi_{\left[\frac{k + 1/2}{2^i}, \frac{k+1}{2^i}\right]}$$
for $i \geq 1, 0 \leq k < 2^i$.
These random variables (known as Haar Functions) have mean zero.  Furthermore, they are uncorrelated.
However, the sum of these random variable do not satisfy the requirements of your theorem because $Var(\sum_{n=1}^\infty X_n) = \infty$.
Here we change our point of view to restate the question as follows: Does there exist an orthonormal basis of $L^2([0,1])$ such that an element written in this basis converges in $L^2$ but does not converge pointwise almost everywhere.  I asked this question in this thread and David Mitra was kind enough to point me towards the paper: Topics in Orthogonal Functions - Price.  On
pg. 598 the author states the Haar functions defined above form a complete orthonormal basis.  Then pg. 603 states

For every complete orthonormal system $\Phi$ there is an $L^2$ function $Y$ whose $\Phi$-Fourier series can be rearranged to diverge almost everywhere.

So we can finish as follows.  Take the Haar functions as above (a complete orthonormal set) and the function $Y$ defined above.  We write this function's $\Phi$-Fourier series as
$$\sum_{n=1}^\infty \langle Y, X_n \rangle X_n$$
Now there exists a rearrangement $\sigma$ of the indices of sum such that it diverges almost everywhere.  Now let $Y_n = \langle Y, X_{\sigma(n)} \rangle X_{\sigma(n)}$ be this rearrangement of terms.  We can see
$$\sum_{n=1}^\infty Y_n$$
satisfies the requirements of the problem.  Indeed
$$\sum_{n=1}^\infty\operatorname{Var} (Y_n) = \operatorname{Var}(Y) < \infty.$$
Furthermore, $\sum Y_n$ diverges almost everywhere.
Note: A fun corollary is that if you have a sequence $X_n$ that does converge almost everywhere (and satisfies the questions requirements), and it forms a complete basis (its already practically orthonormal), then a rearrangement of itself diverges almost everywhere.
