[jason1990]

Okay, I think this is a valid non-anticipating counterexample. Let X_j be 1 or -1 with equal likelihood. Let N be the first time that |Z_N| > 2. By the law of the iterated logarithm, N is finite with probability 1. By symmetry, E[Z_N] = 0. And, of course, E[(Z_N)^2] > 4.

For a bounded example, take min(N,T) for T sufficiently large. This works because |Z_{min(N,T)}| <= |Z_N|, so we can apply dominated convergence.

So it appears that the random session lengths can create bias in either direction. I wonder if one direction is more common than the other and, if so, what would explain that?