Convergence Properties in Certain Occupancy Problems Including the Karlin-Rouault Law
Estáte V. Khmaladze · Journal of Applied Probability · 2011
Letxdenote a vector of lengthqconsisting of 0s and 1s. It can be interpreted as an ‘opinion’ comprised of a particular set of responses to a questionnaire consisting ofqquestions, each having {0, 1}-valued answers. Suppose that the questionnaire is answered bynindividuals, thus providingn‘opinions’. Probabilities of the answer 1 to each question can be, basically, arbitrary and different for different questions. Out of the 2qdifferent opinions, what number, μn, would one expect to see in the sample? How many of these opinions, μn(k), will occur exactlyktimes? In this paper we give an asymptotic expression for μn/ 2qand the limit for the ratios μn(k)/μn, when the number of questionsqincreases along with the sample sizenso thatn= λ2q, where λ is a constant. Letp(x) denote the probability of opinionx. The key step in proving the asymptotic results as indicated is the asymptotic analysis of the joint behaviour of the intensitiesnp(x). For example, one of our results states that, under certain natural conditions, for anyz> 0, ∑1{np(x) >z}=dnz−u,dn=o(2q).