Splitting trees stopped when the first clock rings and Vervaat's transformation

Abstract

We consider a branching population where individuals have independent and identically distributed (i.i.d.) life lengths (not necessarily exponential) and constant birth rates. We let N_t denote the population size at time t. We further assume that all individuals, at their birth times, are equipped with independent exponential clocks with parameter δ. We are interested in the genealogical tree stopped at the first time T when one of these clocks rings. This question has applications in epidemiology, population genetics, ecology, and queueing theory. We show that, conditional on {T<∞}, the joint law of (N_t, T, X^(T)), where X^(T) is the jumping contour process of the tree truncated at time T, is equal to that of (M, -I_M, Y'_M) conditional on {M≠0}. Here M+1 is the number of visits of 0, before some single, independent exponential clock e with parameter δ rings, by some specified Lévy process Y without negative jumps reflected below its supremum; I_M is the infimum of the path Y_M, which in turn is defined as Y killed at its last visit of 0 before e; and Y'_M is the Vervaat transform of Y_M. This identity yields an explanation for the geometric distribution of N_T (see Kitaev (1993) and Trapman and Bootsma (2009)) and has numerous other applications. In particular, conditional on {N_T=n}, and also on {N_T=n,T<a}, the ages and residual lifetimes of the n alive individuals at time T are i.i.d. and independent of n. We provide explicit formulae for this distribution and give a more general application to outbreaks of antibiotic-resistant bacteria in the hospital.