Height and contour processes of Crump-Mode-Jagers forests (I): general distribution and scaling limits in the case of short edges

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 are interested in the height and contour processes encoding a general CMJ tree.We show that the one-dimensional distribution of the height process can be expressed in terms of a random transformation of the ladder height process associated with the underlying Lukasiewicz path. As an application of this result, when edges of the tree are "short" we show that, asymptotically, (1) the height process is obtained by stretching by a constant factor the height process of the associated genealogical Galton-Watson tree, (2) the contour process is obtained from the height process by a constant time change and (3) the CMJ trees converge in the sense of finite-dimensional distributions.

Height and contour processes of Crump-Mode-Jagers forests (II): the Bellman–Harris universality class

“…Beyond the Galton-Watson and Bellman-Harris universality classes treated in [24] and in the present paper, there remains a large class of CMJ forests with "long" edges but where the chronological and genealogical structures remain dependent in the limit (i.e., (IC1) or (IC2) does not hold). In current work in progress, we are looking at the case where P * conditionally on V * is a renewal process stopped at V * : the chronology and the genealogy then remain positively correlated.…”

“…For discrete trees and in [24], the particle is assumed to travel at unit speed along the edges of the forest, in such a way that each point of the tree is visited twice.…”

“…We believe that this is a good choice as far as scaling limits are concerned because of the relations (1.1). In [24], we defined the contour process in the "classical way", i.e., when the exploration particle moves at constant speed. This definition led to a significant number of technical problems due to the absence of tightness of the contour in the Skorohod topology.…”

“…The strong Markov property implies that (T, Z , R) is a trivariate renewal process. With the notation of the present paper, the equation below (2.18) in [24] states that…”

“…Spine process. The spine process was introduced in [24] as an extension of the classical exploration process [19] suited for the chronological setting. Here, we only need a marginal of this process, denoted Π = (Π(n), n ≥ 0), which for simplicity we continue to name spine process.…”