In:
Journal of Applied Probability, Cambridge University Press (CUP), Vol. 50, No. 01 ( 2013-03), p. 208-227
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 & lt;∞}, 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 & lt;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.
Type of Medium:
Online Resource
ISSN:
0021-9002
,
1475-6072
DOI:
10.1017/S0021900200013218
Language:
English
Publisher:
Cambridge University Press (CUP)
Publication Date:
2013
detail.hit.zdb_id:
1474599-9
detail.hit.zdb_id:
219147-7
SSG:
3,2
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