The simple harmonic urn

Abstract

We study a generalized Polya urn model with two types of ball. If the drawn ball is red it is replaced together with a black ball, but if the drawn ball is black it is replaced and a red ball is thrown out of the urn. When only black balls remain, the roles of the colours are swapped and the process restarts. We prove that the resulting Markov chain is transient but that if we throw out a ball every time the colours swap, the process is positive-recurrent. We show that the embedded process obtained by observing the number of balls in the urn at the swapping times has a scaling limit that is essentially the square of a Bessel diffusion. We consider an oriented percolation model naturally associated with the urn process, and obtain detailed information about its structure, showing that the open subgraph is an infinite tree with a single end. We also study a natural continuous-time embedding of the urn process that demonstrates the relation to the simple harmonic oscillator; in this setting our transience result addresses an open problem in the recurrence theory of two-dimensional linear birth and death processes due to Kesten and Hutton. We obtain results on the area swept out by the process. We make use of connections between the urn process and birth-death processes, a uniform renewal process, the Eulerian numbers, and Lamperti's problem on processes with asymptotically small drifts; we prove some new results on some of these classical objects that may be of independent interest. For instance, we give sharp new asymptotics for the first two moments of the counting function of the uniform renewal process. Finally we discuss some related models of independent interest, including a "Poisson earthquakes" Markov chain on the homeomorphisms of the plane

Excursions and path functionals for stochastic processes with asymptotically zero drifts

“…In this section we study a particular Markov chain (A t , B t ) on Z 2 \ {(0, 0)}, with discrete time t ∈ N. The model was introduced in [9], motivated by an urn model. The model takes as input the distribution of a Z-valued random variable κ.…”

“…As well as being key ingredients in the proofs of our large-t asymptotics, these quantities are of interest in their own right in various theoretical and applied contexts. For example, to apply Theorem 2.1 of [10] one needs to understand tail properties of an analogue of η s=1 X α s ; sums over excursions for processes with asymptotically zero drift turn out to be central to the analysis of the 'simple harmonic urn' [9] (see also Section 3.3 below).…”

Impatient Random Walk

“…But first we need the following statement (see e.g. Proposition 7.1 (i)-(iii) in [2]). In particular, when c ∈ [−1/2, 0), X is null recurrent but X imp is positive recurrent whenever α > 1 + 2c (for example, when α > 1, i.e.…”

“…m+1 . Hence, if T + n denotes the total actual time spent on the edges (0, 1), (1,2)...(n − 2, n − 1) before exiting the interval, then…”

“…An interesting phenomenon which arises in our model is that, depending on the passage times, X imp can be positive recurrent even if the original walk was null-recurrent. 2 But one can also imagine that the walker is actually crossing the edge continuously with a speed depending on the passage time sequence. 3 Our original model was more mundane: a person window shopping who gets bored quickly by the stores of any street she has already visited.…”