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

Emmanuel Schertzer,Florian Simatos

doi:10.1214/18-ejp151

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.

Highlights

To generate the height process, we start by labeling the vertices of the tree according to their order of visit by the exploration particle: the height process evaluated at k is given by the distance from the root of the kth vertex
Contour processes of CMJ forests have been considered by Lambert in [14] in the particular setting where birth events are distributed in a Poissonian way along the sticks independently of the life-length – the so-called binary, homogeneous case
In the present paper we prove that if E(Y∗) < ∞, in the nearcritical regime the asymptotic behavior of the chronological height process is obtained by stretching the genealogical height process by the deterministic factor E(Y∗)

Summary

INTRODUCTION

Contour processes of CMJ forests have been considered by Lambert in [14] in the particular setting where birth events are distributed in a Poissonian way along the sticks independently of the life-length – the so-called binary, homogeneous case. In this paper we are interested in the chronological height and contour processes associated to CMJ forests, which corresponds to the case where the planar forest is constructed from an i.i.d. sequence of sticks. This result describes the law of ((T (k) − T (k − 1), Q(k)), k < G) ◦ θn under P and justifies the claim made before the statement of the lemma By combining this result with the spine decomposition of Proposition 3.4, we get that the genealogical height process at a fixed time can be expressed as a functional of an explicit bivariate renewal process.

CONVERGENCE OF THE HEIGHT PROCESS

CONVERGENCE OF THE CONTOUR PROCESS

PRELIMINARY RESULTS

SOME EXAMPLES WHERE TIGHTNESS FAILS