Infinite sum of random variables: subtle convergence question? I have a sequence Xj of random variables, each of which individually is uniformly distributed on the unit circle in the complex plane, and a corresponding sequence cj of positive coefficients. My sequence of coefficients has the property that $\sum_{j=1}^\infty c_j^2$ converges but $\sum_{j=1}^\infty c_j$ diverges.
Note that I am not assuming that the Xj are independent. What I do know about the variables is this: any odd-index one $X_{2j-1}$ is independent from any finite collection of other $X_i$. In other words, all the odd-index ones are independent of one another and of the even-index ones, but there might be dependences among the even-index ones.
I want to write down three other random variables:
$$
A = \sum_{j=1}^\infty c_j X_j, \quad
B_1 = \sum_{j=1}^\infty c_{2j-1} X_{2j-1}, \quad
B_2 = \sum_{j=1}^\infty c_{2j} X_{2j}.
$$
That B1 is a well-defined random variable is no problem: since the X2j-1 are all independent, the sum defining B1 exists almost surely thanks to the $\ell^2$-convergence of the cj.
There is no immediate reason to think that A itself is a well-defined random variable; however, in my situation, I have extra information that ensures that A really is well-defined.
So my two questions (finally) are these:


*

*Is the above information enough to prove that B2 is a well-defined random variable?

*Is the above information enough to prove that B1 and B2 are independent?

 A: For the First one I don't know sorry.
For the second one the answer is Yes 
You might consider the Sigma-algebra generated by your family $X_{2j-1}$ and the see that it is independent from the Sigma-algebra generated by your family $X_{2j}$ by recurrence.
I can add that if you cannot see it you can have a look at the 0's Chapter of Itô's lectures at Aarhus University viewable here 
Regards
A: To get around the worry about using properties of the $X_{2j}$ without a priori convergence of the series defining $B_2$, you might use a relative of the following result.
If the $\pi$-systems of events $H_{\alpha,\beta}$, $(\alpha,\beta) \in \mathcal{J}$, are mutually independent, then the $\sigma$-algebras $$G_\alpha =\sigma \left(\bigcup_\beta H_{\alpha,\beta} \right)$$ are mutually independent.
I got this out of Amir Dembo's notes, but I imagine the result is in many books (see, e.g. Durrett's Probability: Theory and Examples, Third Edition, Chapter 1, (4,4).)
To apply the proposition, we would let $\alpha$ run over $1,2$. Let $H_{1,\beta}=\sigma(X_{2 \beta-1})$ and let $H_{2,\beta} = \sigma(X_{2\beta})$. The $\beta$, of course, range over the natural numbers. Unfortunately, the hypothesis doesn't quite hold due to the dependence of the $X_{2j}$.
But perhaps it might help to look at the proof and see if we can weaken the assumption of mutual independence of all the $H_{\alpha,\beta}$? Essentially we might define $K_{\alpha,\ell} = \sigma(H_{\alpha,1},H_{\alpha,2},\ldots,H_{\alpha,\ell})$ and assume $K_{1,\ell}$ is independent of $K_{2,m}$ for every $m$ and $n$. Can we conclude then then the limits are independent by reasoning similar to that of the above proposition? I suspect not exactly -- we need some finer control than that, but maybe it can be extended...
