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 (pj) j≥1 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 ν(x) = |{j: pj ≥ x}|. In a regularly varying urn scheme, ν satisfies limτ→0 ν(τ x)/ ν(τ ) = x −α for α ∈ [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.

Functional central limit theorems for certain statistics in an infinite urn scheme

“…This paper extends the results of [7] and [6], where a functional central limit theorem (FCLT) was shown under condition (3) for the vector process 1] in the case θ ∈ (0, 1]. Ordinary (not functional) central limit theorems for the above quantities were established under various conditions in [2], [3], [9], [10], [12], [13], [14]. In particular, under rather general conditions on the sequence (p i ) involving an unbounded growth of the variances, the following results available: a strong law of large numbers and asymptotic normality of R n , an asymptotic normality of the vector (R n,1 , .…”

On nested infinite occupancy scheme in random environment

“…Our main result, Theorem 2.1, is an immediate consequence of Proposition 3.7 given in Section 3.2, Theorem 3.2 given next and its corollary. (8) and (9). Then…”

“…(b) Condition(8) ensures that Var N (t) = O(t 2γ ) as t → ∞. Pick any δ > 0 such that δ(ω − γ) > 1/2.…”

A Good-Turing estimator for feature allocation models

“…wherem n ,v − n andv + n are suitable quantities that will be defined in the proof. First we discuss how to determine (18), we remind that D n =M n − M n is a sub-Gamma random variable, as shown in Proposition A.2, hence the following holds (see Ben-Hamou et al (2017)) P(D n > E(D n ) + 2v + n s + s/n) ≤ e −s for any s ≥ 0. Putting e −s = δ, we obtain P(D n ≤ E(D n ) + 2v + n log(1/δ) + log(1/δ)/n) ≥ 1 − δ.…”