Convergence Properties in Certain Occupancy Problems Including the Karlin-Rouault Law

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 ).

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