Almost sure and moment convergence for triangular P\'olya urns

We consider triangular P\'olya urns and show under very weak conditions a general strong limit theorem of the form $X_{ni}/a_{ni}\to \mathcal{X}_i$ a.s., where $X_{ni}$ is the number of balls of colour $i$ after $n$ draws; the constants $a_{ni}$ are explicit and of the form $n^\alpha\log^\gamma n$; the limit is a.s. positive, and may be either deterministic or random, but is in general unknown. The result extends to urns with subtractions under weak conditions, but a counterexample shows that some conditions are needed. For balanced urns we also prove moment convergence in the main results if the replacements have the corresponding moments. The proofs are based on studying the corresponding continuous-time urn using martingale methods, and showing corresponding results there. We assume for convenience that all replacements have finite second moments.