Abstract
Crump-Mode-Jagers (CMJ) trees generalize Galton-Watson trees by allowing individuals to live for an arbitrary duration and give birth at arbitrary times during their life-time. In this paper, we exhibit a simple condition under which the height and contour processes of CMJ forests belong to the universality class of Bellman-Harris processes. This condition formalizes an asymptotic independence between the chronological and genealogical structures. We show that it is satisfied by a large class of CMJ processes and in particular, quite surprisingly, by CMJ processes with a finite variance offspring distribution. Along the way, we prove a general tightness result.
Highlights
Crump–Mode–Jagers (CMJ) trees generalize Galton–Watson trees by allowing individuals to live for an arbitrary duration and give birth at arbitrary times during their life-time
1.1 Crump-Mode-Jagers forests The subject of the present paper is the study of the height and contour processes of planar Crump–Mode–Jagers (CMJ) forests, which are random instances of chronological forests
The chronological tree corresponding to a Bellman–Harris process can be obtained by putting i.i.d. marks on the edges of the corresponding genealogical tree, that can be seen as stretching factors
Summary
1.1 Crump-Mode-Jagers forests The subject of the present paper is the study of the height and contour processes of planar Crump–Mode–Jagers (CMJ) forests, which are random instances of chronological forests. As noted by Lambert [17], a chronological tree can be regarded as a tree satisfying the rule “edges always grow to the right” We define the chronological height process at time n, denoted H(n), as the date of birth of the nth individual in the forest. The chronological height can be obtained by summing up the “chronological contribution” of each ancestor along the ancestral line associated with n To formalize this statement, we consider the spine at time n, denoted Π(n), which is the measure recording each of those contributions along the spine.
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