For balanced Polya urns, the p-th moments of centered ball counts obey explicit polynomial-in-n bounds matching the urn's dominant eigenvalues, yielding moment convergence whenever a central limit theorem is known.
Bernstein: Sur un probl` eme du sch´ ema des urnes ` a compo- sition variable
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Moments of balanced P\'olya urns
For balanced Polya urns, the p-th moments of centered ball counts obey explicit polynomial-in-n bounds matching the urn's dominant eigenvalues, yielding moment convergence whenever a central limit theorem is known.