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The collision spectrum of \(\Lambda\)-coalescents. (English) Zbl 1404.60104

Summary: \(\Lambda\)-coalescents model the evolution of a coalescing system in which any number of blocks randomly sampled from the whole may merge into a larger block. For the coalescent restricted to initially \(n\) singletons, we study the collision spectrum \((X_{n,k}:2\leq k\leq n)\), where \(X_{n,k}\) counts, throughout the history of the process, the number of collisions involving exactly \(k\) blocks. Our focus is on the large \(n\) asymptotics of the joint distribution of the \(X_{n,k}\)’s, as well as on functional limits for the bulk of the spectrum for simple coalescents. Similar to the previous studies of the total number of collisions, the asymptotics of the collision spectrum largely depends on the behaviour of the measure \(\Lambda\) in the vicinity of \(0\). In particular, for beta\((a,b)\)-coalescents different types of limit distributions occur depending on whether \(0<a\leq1\), \(1<a<2\), \(a=2\) or \(a>2\).

MSC:

60J25 Continuous-time Markov processes on general state spaces
60F17 Functional limit theorems; invariance principles
60C05 Combinatorial probability
60G09 Exchangeability for stochastic processes
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