Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws

Abstract

This paper collects facts about the number of occupied boxes in the classical balls-in-boxes occupancy scheme with infinitely many positive frequencies: equivalently, about the number of species represented in samples from populations with infinitely many species. We present moments of this random variable, discuss asymptotic relations among them and with related random variables, and draw connections with regular variation, which appears in various manifestations.

Published at http://dx.doi.org/10.1214/07-PS092 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (23)

• Languages cool as they expand: Allometric scaling and the decreasing need for new words
• Sparse graphs using exchangeable random measures
• Beta-coalescents and continuous stable random trees
• Scaling laws and fluctuations in the statistics of word frequencies
• Local limit theorems for finite and infinite urn models
• Concentration inequalities in the infinite urn scheme for occupancy counts and the missing mass, with applications
• Dense Power-law Networks and Simplicial Complexes
• On the number of collisions in beta(2, $b$)-coalescents
• On the variance of the number of occupied boxes
• Stochastic approximations to the Pitman-Yor process
• The Bernoulli sieve revisited
• Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process
• On sparsity, power-law and clustering properties of graphex processes
• About adaptive coding on countable alphabets
• On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I
• A Berry-Esseen bound for the uniform multinomial occupancy model
• On the occupancy problem for a regime switching model
• Gaps and interleaving of point processes in sampling from a residual allocation model
• Asymptotic Analysis of Statistical Estimators related to MultiGraphex Processes under Misspecification
• Laws of the iterated and single logarithm for sums of independent indicators, with applications to the Ginibre point process and Karlin's occupancy scheme
• Records in the Infinite Occupancy Scheme
• The Normal-Generalised Gamma-Pareto process: A novel pure-jump Lévy process with flexible tail and jump-activity properties
• The Minimax Risk in Testing Uniformity over Large Alphabets under Missing-Ball Alternatives