A.s. convergence for infinite colour Pólya urns associated with random walks

Abstract

We answer Problem 11.1 of Janson [6] on Pólya urns associated with stable random walk. Our proof use neither martingales nor trees, but an approximation with a differential equation.Acknowledgment I am supported by the ERC consolidator grant 101001124 (UniversalMap).

Strong Convergence of Infinite Color Balanced Urns Under Uniform Ergodicity

“…The result essentially completes the work developed in [4,24] for the case when S is countable. We would like to note here that similar results for a null recurrent case (when the chain is a random walk) has been derived in [24,20].…”

“…A new generalization for balanced urn schemes with infinitely many colors was again introduced in [29] and subsequently in the papers [3,4,5]. These work have since then generated a lot of interest and such models are now receiving considerable attention [20,19,24]. In this paper, we will consider the infinite color balanced urn model, where the color set is countably infinite.…”

“…For countable many colors, we prove almost sure convergence of the urn configuration under uniform ergodicity assumption on the associated Markov chain. The proof uses a stochastic coupling of the sequence of chosen colors with a branching Markov chain on a weighted random recursive tree as described in [5,20]. Using this coupling we estimate the covariance between any two selected colors.…”

Predictive Constructions Based on Measure-Valued Pólya Urn Processes

“…There, the class of randomly reinforced urns [17] assumes an R with zero off-diagonal elements (i.e., we reinforce only the color of the observed ball), whereas the generalized Pólya urn models require the mean replacement matrix to be irreducible. Similarly to the k-color case, RRPPs need the use of different techniques, which yield completely different results than those in [8,9,[14][15][16]. As an example, Theorem 1 in [16] and our Theorem 2 prove convergence of the kind (2), yet the limit probability measure in [16] is non-random.…”

“…Existing studies on MVPPs look at models that have mostly a balanced design, i.e., R x (X) = r, x ∈ X, and assume irreducibility-like conditions for (R x ) x∈X , see [8,9,14,15] and Remark 4 in [16]. In contrast, RRPPs require that R x ({x} c ) = 0, and so are excluded from the analysis in those papers.…”