Bounded size biased couplings, log concave distributions and concentration of measure for occupancy models

Abstract

Threshold-type counts based on multivariate occupancy models with log concave marginals admit bounded size biased couplings under weak conditions, leading to new concentration of measure results for random graphs, germ-grain models in stochastic geometry, multinomial allocation models and multivariate hypergeometric sampling. The work generalizes and improves upon previous results in a number of directions.

35 pages. Version 2 is the result of a major revision, including corrections and significant clarifications in particular as regards Lemmas 2.6 and 2.7, and the introduction and use of Property (B,*) in Definition 2.2. Version 3 added some references and removed Theorem 1.1