A Pólya urn is balanced when each draw changes the total number of balls by the same amount regardless of which color is drawn. This balance makes the urn tractable — the total ball count grows deterministically, and the color proportions converge to a well-characterized limit.
Desmarais relaxes this: the urn need only be balanced in expectation. Each draw's contribution to the total count can be random, but its expected value must be constant across colors. The individual steps fluctuate; the aggregate growth rate does not.
This weaker condition still yields a normal limit law with convergence of all moments — not just the first two, but every moment of the distribution converges to the Gaussian target. The applications demonstrate the generalization's reach: degree distributions in random trees with uniform attachment and freezing (where vertices can stop receiving edges), and hooking networks (where edges connect components through random hooks). Both models produce urn dynamics that are unbalanced step-by-step but balanced on average.
The structural point: the classical theory demanded deterministic balance at every step. The extension shows that balance only needs to hold in expectation — the randomness in individual steps washes out over the aggregate. This is a precise instance of the broader principle that stochastic systems often inherit the limit behavior of their deterministic skeletons, provided the noise averages to zero. The fluctuations are real but irrelevant to the limit.