A New Approach to Pólya Urn Schemes and Its Infinite Color Generalization

Random replacements in Pólya urns with infinitely many colours

“…The purpose of the present note is to show that this model with a measurevalued Pólya urn and the results for it by [3] and [12] extend almost automatically to the case of random replacements, at least in the case with no removals. In fact, we show that the model is so flexible that a random replacement can be seen as a deterministic replacement using the larger colour space S × [0, 1], where the extra coordinate is used to simulate the randomization.…”

“…(The extensions are all usually called Pólya urns, or perhaps generalized Pólya urns.) These generalizations have been studied by a large number of authors, and have found a large number of applications, see for example [11], [8], [3], [12] and the references given there. The extensions include (but are not limited to) the following, in arbitrary combinations.…”

“…One example is Bandyopadhyay and Thacker [2,1] who studied the case when the space of colours is Z d , and the replacements are translation invariant. A very general version of Pólya urns was introduced by Blackwell and MacQueen [4] in a special case (with every replacement having the colour of the drawn ball, as in the original Pólya urn, see Example 3.2), and much more generally (with rather arbitrary deterministic replacements) by Bandyopadhyay and Thacker [3] and Mailler and Marckert [12]; this version can be described by: (vi) The space S of colours is a measurable space. The state space is now the space M(S) of finite measures on S; if the current state is µ, then the next ball is drawn with the distribution µ/µ(S).…”

“…Remark 1.2. The colour space S is assumed to be a Polish topological space in [4] and [12], and for the convergence results in [3], while the representation results in [3] are stated for a general S. We too make our definitions for an arbitrary measurable space S, but we restrict to Borel spaces in our main result. (This includes the case of a Polish space, see Lemma 2.1 below.…”

“…We include there a detailed treatment of measurability questions, showing that there are no such problems. (This was omitted in [3] and [12], where the situation is simpler and straightforward. In our, technically more complex, situation, there is a need to verify measurability explicitly.…”