The CLT Analogue for Cyclic Urns

Abstract

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.MSC2010: 60F05, 60F15, 60C05, 60J10.

Two-color balanced affine urn models with multiple drawings

“…We also refer to the classic surveys of Johnson and Kotz [13,19], the book of Mahmoud [23] and the references therein. The recent works of Chauvin et al [2,3], Janson [10,11,12], Neininger and Knape [17], Pouyanne [29], Mailler [25], M üller and Neininger [27], M üller [28], are all devoted to urn models where only a single ball is sampled at each step.…”

Refined asymptotics for the composition of cyclic urns

“…Here and subsequently, we make no use of the fact that the martingale limits Ξ k can also be written explicitly as deterministic functions of the limit of the random binary search tree when interpreting the evolution of the random binary search tree as a transient Markov chain and its limit as a random variable in the Markov chain's Doob-Martin boundary, see [6,8]. Following this path the Ξ k become a deterministic function of (U n ) n≥1 and from this representation the self-similarity relation (15) can be read off as well. See [1] for a related explicit construction.…”

On densities for solutions to stochastic fixed point equations

“…Over the last 25 years, this approach has been extended to a variety of random variables with underlying recursive structures. Some examples are recursive algorithms , data structures , Pólya urn models , and random tree models .…”

“…Over the last 25 years, this approach has been extended to a variety of random variables with underlying recursive structures. Some examples are recursive algorithms [29,31,34,38], data structures [21,31,32], Pólya urn models [18,27], and random tree models [1,20]. Limit distributions derived by the contraction method are given implicitly as solutions to stochastic fixed point equations.…”