Branching trees I: Concatenation and infinite divisibility

Beitrag in einer Fachzeitschrift
(Online-Publikation)


Details zur Publikation

Autorinnen und Autoren: Glöde PK, Greven A, Rippl T
Zeitschrift: arXiv
Jahr der Veröffentlichung: 2017
ISSN: 2331-8442
Sprache: Englisch


Abstract

The goal of this work is to decompose random populations with a genealogy in
subfamilies of a given degree of kinship and to obtain a notion of infinitely divisible
genealogies. We model the genealogical structure of a population by (equivalence
classes of) ultrametric measure spaces (um-spaces) as elements of the Polish space
U which we recall. In order to then analyze the family structure in this coding we
introduce an algebraic structure on um-spaces (a consistent collection of semigroups).
This allows us to obtain a path of collections of subfamilies of fixed kinship h (described
as ultrametric measure spaces), for every depth h as a measurable functional of the
genealogy.
Technically the elements of the semigroup are those um-spaces which have diam-
eter less or equal to 2h called h-forests (h > 0). They arise from a given ultrametric
measure space by applying maps called h−truncation. We can define a concatena-
tion of two h-forests as binary operation. The corresponding semigroup is a Delphic
semigroup and any h-forest has a unique prime factorization in h-trees (um-spaces
of diameter less than 2h). Therefore we have a nested R + -indexed consistent (they
arise successively by truncation) collection of Delphic semigroups with unique prime
factorization.
Random elements in the semigroup are studied, in particular infinitely divisible
random variables. Here we define infinite divisibility of random genealogies as the
property that the h-tops can be represented as concatenation of independent identi-
cally distributed h-forests for every h and obtain a Lévy-Khintchine representation
of this object and a corresponding representation via a concatenation of points of a
Poisson point process of h-forests.
Finally the case of discrete and marked um-spaces is treated allowing to apply the
results to both the individual based and most important spatial populations.
The results have various applications. In particular the case of the genealogical
(U-valued) Feller diffusion and genealogical (U V -valued) super random walk is treated
based on the present work in [DG18b] and [GRG].
In the part II of this paper we go in a different direction and refine the study in
the case of continuum branching populations, give a refined analysis of the Laplace
functional and give a representation in terms of a Cox process on h-trees, rather than
forests.


FAU-Autorinnen und Autoren / FAU-Herausgeberinnen und Herausgeber

Glöde, Patric Karl
Department Mathematik
Greven, Andreas Prof. Dr.
Lehrstuhl für Mathematische Stochastik
Rippl, Thomas Dr.
Lehrstuhl für Mathematische Stochastik


Zitierweisen

APA:
Glöde, P.K., Greven, A., & Rippl, T. (2017). Branching trees I: Concatenation and infinite divisibility. arXiv.

MLA:
Glöde, Patric Karl, Andreas Greven, and Thomas Rippl. "Branching trees I: Concatenation and infinite divisibility." arXiv (2017).

BibTeX: 

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