Refined asymptotics for the composition of cyclic urns

Abstract

A cyclic urn is an urn model for balls of types 0, . . . , m − 1. The urn starts at time zero with an initial configuration. Then, in each time step, first a ball is drawn from the urn uniformly and independently from the past. If its type is j, it is then returned to the urn together with a new ball of type j + 1 mod m. The case m = 2 is the well-known Friedman urn. The composition vector, i.e., the vector of the numbers of balls of each type after n steps is, after normalization, known to be asymptotically normal for 2 ≤ m ≤ 6. For m ≥ 7 the normalized composition vector is known not to converge. However, there is an almost sure approximation by a periodic random vector.In the present paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all 7 ≤ m ≤ 12. For m ≥ 13 we also find asymptotically normal fluctuations when normalizing in a more refined way. These fluctuations are of maximal dimension m − 1 only when 6 does not divide m. For m being a multiple of 6 the fluctuations are supported by a two-dimensional subspace.MSC2010: 60F05, 60F15, 60C05, 60J10.

Central limit theorem analogues for multicolour urn models

“…On the other hand, if there is more than one eigenvalue with real part greater than r/2, the asymptotic expansion of the urn composition typically contains random terms of size larger than √ n which may even oscillate, compare [8,10,32]. In such a situation, the fluctuation about these random tendencies is of some interest [22,30]. This question is addressed in the present article, and in order to study the fluctuations, we employ a "non-classical" normalisation of the urn composition vector that involves random centering and possibly random scaling.…”

On rates of convergence in the probabilistic analysis of algorithms

“…For the proof of Theorem 5.1, we need a slightly stronger version of this bound which also applies if the appearing quantities are dependent. The following statement can be found in Müller and Neininger [66,Lemma 3.4] (see also Neininger [68, Lemma 2.1] for a similar statement for the univariate case).…”

“…n ) n≥0 and the toll term b n show dependencies (see Neininger [68] for refined Quicksort asymptotics or Müller and Neininger [66] for the composition of cyclic urns). Therefore, the aim of this chapter is to derive a general convergence theorem similar to Theorem 3.4, with the difference that we replace the conditional independence condition (1.3) by the (weaker) partial conditional independence condition (1.4), i.e., by the assumption that (A 1 (n), .…”

“…As already mentioned, possible examples of distributional recurrences with dependent toll function start with the refined convergence results for Quicksort in Neininger [68] and for cyclic urns in Müller and Neininger [66], see also Fuchs, Müller and Sulzbach [32]. Since [68] only contains a convergence result for the Zolotarev metric without a rate of convergence, we take up this example in Section 5.3.1 and use Theorem 5.1 to estimate the rate of convergence.…”

“…For all these examples, the contraction method can be used to derive distributional limit laws. However, it turns out that in some special cases of normal limits, the contraction method can be applied although the conditional independence condition (1.3) is not satisfied, see Neininger [68] and Müller and Neininger [66]. In order to also cover similar applications, we formulate a slightly weakened independence condition:…”