Convergence properties in certain occupancy problems including the Karlin-Rouault law

Abstract

Let x denote a vector of length q consisting of 0s and 1s. It can be interpreted as an `opinion' comprised of a particular set of responses to a questionnaire consisting of q questions, each having {0, 1}-valued answers. Suppose that the questionnaire is answered by n individuals, thus providing n `opinions'. Probabilities of the answer 1 to each question can be, basically, arbitrary and different for different questions. Out of the 2^q different opinions, what number, μ_n, would one expect to see in the sample? How many of these opinions, μ_n(k), will occur exactly k times? In this paper we give an asymptotic expression for μ_n / 2^q and the limit for the ratios μ_n(k)/μ_n, when the number of questions q increases along with the sample size n so that n = λ2^q, where λ is a constant. Let p(x) denote the probability of opinion x. The key step in proving the asymptotic results as indicated is the asymptotic analysis of the joint behaviour of the intensities np(x). For example, one of our results states that, under certain natural conditions, for any z > 0, ∑1_{{np(x) > z}} = d_n z^-u, d_n = o(2^q).