2015/06/10 by Emmanuel Schertzer, Schertzer, Emmanuel, Florian Simatos +1 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR
paper · pdf · doi:10.48550/arxiv.1506.03192
introduction of the spine process, generalizing to the chronological setting the classical discrete exploration process
openalex publication_date 2015/06/10 · arxiv created 2016/07/27 · arxiv updated 2016/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.