An urn model for cannibal behavior

Abstract

A sampling procedure involving an urn with red and white balls in it is studied. Initially, the urn contains n balls, r of them being white. At each step, a white ball is removed, and one more ball is selected at random, painted red (if it was white before) and put back into the urn. R. F. Green proposed this scheme in 1980 as a stochastic model of cannibalistic behavior in a biological population, with red balls interpreted as cannibals. Of primary interest is the distribution of Xnr, the terminal number of red balls. A study of R. F. Green and C. A. Robertson led them to conjecture that, for r = 1 and n →∞, Xnr is asymptotically normal with mean ≈ n exp(–1) and variance ≈ n(3 exp(–2) –exp(–1)). In this paper we prove that the conjecture — its natural extension, in fact — is true. Namely, for r/n bounded away from 1, Xnr is shown to be asymptotically normal with mean ≈ n exp(ρ – 1) and variance ≈ n exp[2(ρ – 1)] (ρ2– 3ρ + 3 – exp(l – ρ)); ρ = r/n.