The rate of convergence of the block counting process of exchangeable coalescents with dust

For exchangeable coalescents with dust the rate of convergence as the sample size tends to infinity of the scaled block counting process to the frequency of singleton process is determined. This rate is expressed in terms of a certain Bernstein function. The proofs are based on Taylor expansions of the infinitesimal generators and semigroups and involve a particular concentration inequality arising in the context of Karlin’s infinite urn model.