Concentration inequalities in the infinite urn scheme for occupancy counts and the missing mass, with applications

Abstract

An infinite urn scheme is defined by a probability mass function $(p_j)_{j\geq1}$ over positive integers. A random allocation consists of a sample of $N$ independent drawings according to this probability distribution where $N$ may be deterministic or Poisson-distributed. This paper is concerned with occupancy counts, that is with the number of symbols with $r$ or at least $r$ occurrences in the sample, and with the missing mass that is the total probability of all symbols that do not occur in the sample. Without any further assumption on the sampling distribution, these random quantities are shown to satisfy Bernstein-type concentration inequalities. The variance factors in these concentration inequalities are shown to be tight if the sampling distribution satisfies a regular variation property. This regular variation property reads as follows. Let the number of symbols with probability larger than $x$ be $\vec{\nu}(x)=|\{j:p_j\geq x\}|$. In a regularly varying urn scheme, $\vec{\nu}$ satisfies $\lim_{\tau \rightarrow0}\vec{\nu}(\tau x)/\vec{\nu}(\tau)=x^{-\alpha}$ for $\alpha\in[0,1]$ and the variance of the number of distinct symbols in a sample tends to infinity as the sample size tends to infinity. Among other applications, these concentration inequalities allow us to derive tight confidence intervals for the Good--Turing estimator of the missing mass.