2015/07/29 by Noela S. Müller, Müller, Noela S., Ralph Neininger +1
Computer Science · Mathematics · #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Probability (math.PR) #cs.DM #math.PR
paper · pdf · doi:10.48550/arxiv.1507.08119
Extended abstract to be replaced later by a full version
arxiv created 2015/07/29 · arxiv updated 2015/07/30
A cyclic urn is an urn model for balls of types 0,…,m-1 where in each draw the ball drawn, say of type j, is 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 does not converge. However, there is an almost sure approximation by a periodic random vector. In this paper the asymptotic fluctuations around this periodic random vector are identified. We show that these fluctuations are asymptotically normal for all m≥ 7. However, they 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.