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 ).

On the asymptotic properties of a certain class of goodness-of-fit tests associated with multinomial distributions

“…where I{}  denotes the indicator function, which are respectively, the number of intervals consisting exactly r and at least r observations, and the number of collisions (that is, the number of observations that we observe in intervals already containing observations). These CS have been considered in the literature in various contexts; see, for instance, Kolchin et al (1976), L'ecuyer et al (2002), Khmaladze (2011).…”

Correction The probabilities of large deviations for a certain class of statistics associated with multinomial distributions

“…r.v.s.. PDS, specifically statistics (1.2), play a leading role in goodness-of-fit tests on grouped data, whereas CS (1.3), (1.4) are used in problems associated with occupancy problems, see Kolchin et al (1976), Ivanov V.A. (1985), Khmaladze (2011). We also draw the attention of readers to the article by L'ecuyer et al (2002), where the statistics (1.1) and its special versions, including (1.2) and (1.3) are used in construction the serial tests for testing of uniformity and independence of the output sequence of general-purpose uniform random number generators.…”

Asymptotically Normal Estimators for Zipf’s Law

“…Processes {J i (Π(t)) def = Π i (t), t ≥ 0} are independent Poisson with parameters p i . Along with the listed papers, the poissonization was used by Ben-Hamou, Boucheron & Gassiat (2016) in estimating codes on countable alphabets, by Durieu & Wang (2016) for proof of functional CLT for some randomization of statistics R n and U n , by Grubel & Hitczenko (2009) in studying limit distributions of gaps in discrete random samples, by Khmaladze (2011) for more general occupancy schemes.…”