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?