2017/01/02 by Götz Kersting, Kersting, Götz, Jason Schweinsberg +3
Computer Science · Mathematics · #60J27 #60J75 #60K20 #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1701.00549
openalex publication_date 2017/01/02 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We consider the number of blocks involved in the last merger of a Λ-coalescent started with n blocks. We give conditions under which, as n → ∞, the sequence of these random variables a) is tight, b) converges in distribution to a finite random variable or c) converges to infinity in probability. Our conditions are optimal for Λ-coalescents that have a dust component. For general Λ, we relate the three cases to the existence, uniqueness and non-existence of quasi-invariant measures for the dynamics of the block-counting process, and in case b) investigate the time-reversal of the block-counting process back from the time of the last merger.