I have a sequence X_{j} of random variables, each of which individually is uniformly distributed on the unit circle in the complex plane, and a corresponding sequence c_{j} 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 X_{j} 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 B_{1} is a well-defined random variable is no problem: since the X_{2j-1} are all independent, the sum defining B_{1} exists almost surely thanks to the $\ell^2$-convergence of the c_{j}.

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 B
_{2}is a well-defined random variable? - Is the above information enough to prove that B
_{1}and B_{2}are independent?