Limit theorems for renewal shot noise processes with decreasing response function

A. Iksanov, A. Marynych, M. Meiners

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We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1; b) measure converges in distribution to a 1-stable law, as the initial number of particles goes to infinity. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance b = 1, which corresponds to the Bolthausen-Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta (a; b)-coalescents with 0 <a <1 leads to a simplied derivation of the known (2 - a)-stable limit. We furthermore derive asymptotic expansions for the (centered) moments of the number of collisions and of the total branch length for the beta (1; b)-coalescent by exploiting the method of sequential approximations.
Originele taal-2Engels
Plaats van productieEindhoven
UitgeverijEurandom
Aantal pagina's58
StatusGepubliceerd - 2012

Publicatie series

NaamReport Eurandom
Volume2012023
ISSN van geprinte versie1389-2355

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