Refined asymptotics for the composition of cyclic urns

Noela Müller,Ralph Neininger

doi:10.1214/18-ejp243

Abstract

A cyclic urn is an urn model for balls of types $0,\ldots ,m-1$. 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 $2\le m\le 6$. For $m\ge 7$ 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 $7\le m\le 12$. For $m\ge 13$ 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.

Highlights

Introduction and resultA cyclic urn is an urn model with a fixed number m ≥ 2 of possible colors of balls which we call types 0, . . . , m − 1
We denote by recurrence for the sequence (Rn) = (Rn,0, . . . , Rn,m−1)t the vector of the numbers of balls of each type after n steps when starting with one ball of type 0
In the present paper we study the fluctuations of n−λ1

Summary

Introduction and result

A cyclic urn is an urn model with a fixed number m ≥ 2 of possible colors of balls which we call types 0, . . . , m − 1. M − 1} it is placed back to the urn together with a new ball of type j + 1 mod m For m ∈ {7, 8, 9, 10, 11} Theorem 1.1 shows that there is a direct normalization of the residuals which implies a multivariate central limit law (CLT). Theorems 1.1 and 1.2 describe refined residuals which satisfy a multivariate CLT for all m > 12. These can be considered as asymptotic expansions of the random variables Rn. The convergences in Theorems 1.1 and 1.2 hold for all moments. The remainder of the present paper contains a proof of Theorems 1.1 and 1.2. The results of this paper were announced in the extended abstract [15]

Explanation of the result and outline of the proof

Convergence of the covariance matrix

Embedding and recursions

The Zolotarev metric

Preparatory lemmata