Branching trees I: concatenation and infinite divisibility (original) (raw)

Branching Processes - A General Concept

Latin American Journal of Probability and Mathematical Statistics, 2021

The paper has four goals. First, we want to generalize the classical concept of the branching property so that it becomes applicable for historical and genealogical processes (using the coding of genealogies by ($V$-marked) ultrametric measure spaces leading to state spaces mathbbU\mathbb{U}mathbbU resp. mathbbUV\mathbb{U}^VmathbbUV). The processes are defined by well-posed martingale problems. In particular we want to complement the corresponding concept of infinite divisibility developed in \cite{infdiv} for this context. Second one of the two main points, we want to find a corresponding characterization of the generators of branching processes more precisely their martingale problems which is both easy to apply and general enough to cover a wide range of state spaces. As a third goal we want to obtain the branching property of the mathbbU\mathbb{U}mathbbU-valued Feller diffusion respectively mathbbUV\mathbb{U}^VmathbbUV-valued super random walk and the historical process on countable geographic spaces the latter as two examples of...

Distributions and measures on the boundary of a tree

Journal of Mathematical Analysis and Applications, 2004

In this paper, we analyze the space D of distributions on the boundary Ω of a tree and its subspace B 0 , which was introduced in [Amer. J. Math. 124 (2002) 999-1043] in the homogeneous case for the purpose of studying the boundary behavior of polyharmonic functions. We show that if µ ∈ B 0 , then µ is a measure which is absolutely continuous with respect to the natural probability measure λ on Ω, but on the other hand there are measures absolutely continuous with respect to λ which are not in B 0. We then give the definition of an absolutely summable distribution and prove that a distribution can be extended to a complex measure on the Borel sets of Ω if and only if it is absolutely summable. This is also equivalent to the condition that the distribution have finite total variation. Finally, we show that for a distribution µ, Ω decomposes into two subspaces. On one of them, a union of intervals A µ , µ restricted to any finite union of intervals extends to a complex measure and on A µ we give a version of the Jordan, Hahn, and Lebesgue-Radon-Nikodym decomposition theorems. We also show that there is no interval in the complement of A µ in which any type of decomposition theorem is possible. All the results in this article can be generalized to results on good (in particular, compact infinite) ultrametric spaces, for example, on the p-adic integers and the p-adic rationals.

On the genealogy on conditioned stable L\'evy forest

Arxiv preprint arXiv:0706.2605, 2007

We give a realization of the stable Lévy forest of a given size conditioned by its mass from the path of the unconditioned forest. Then, we prove an invariance principle for this conditioned forest by considering k independent Galton-Watson trees whose offspring distribution is in the domain of attraction of any stable law conditioned on their total progeny to be equal to n. We prove that when n and k tend towards +∞, under suitable rescaling, the associated coding random walk, the contour and height processes converge in law on the Skorokhod space respectively towards the "first passage bridge" of a stable Lévy process with no negative jumps and its height process.

Genealogy-valued Feller diffusion

arXiv (Cornell University), 2019

We consider the evolution of the genealogy of the population currently alive in a Feller branching diffusion. In contrast to the approach via labeled trees in the continuum random tree world [Ald91a, LG93], following [GPW13], the genealogies are modelled as elements of a Polish space Í which consists of all equivalence classes of ultrametric measure spaces. This space equipped with an operation called concatenation, denoted by (Í, ⊔) has a rich algebraic (semigroup) structure, [GGR19, GRG21], which is used effectively to study branching processes. We focus on the evolution of the genealogy in time and the large time asymptotics conditioned on survival up to present time and on survival forever. We develop the calculus in such a way that it can be applied in the future to more complicated systems, such as logistic branching or state dependent branching. Furthermore the approach we take carries over very smoothly to spatial models with infinitely many components. We prove existence, uniqueness, continuity of paths and a generalized Feller property of solutions of the martingale problem for this genealogy-valued, i.e. Í-valued Feller diffusion. The uniqueness is shown via Feynman-Kac duality with the distance matrix augmented Kingman coalescent. By conditioning on the entire population size process and then observing the genealogy part we obtain the precise relation to a specific time-inhomogeneous Í 1-valued Fleming-Viot process with varying resampling rate, Í 1 being the set of all equivalence classes of ultrametric probability measure spaces. This relation gives the so-called skew martingale representation of the Í-valued Feller diffusion. Via the Feynman-Kac duality we deduce the generalized branching property of the Í-valued Feller diffusion. Using a semigroup operation through concatenations on Í, [GGR19], together with the generalized branching property, [GRG21], we obtain a Lévy-Khintchine formula for the Í-valued Feller diffusion and determine explicitly the Lévy measure which has a special form, allowing us to obtain for h > 0 a decomposition into depth-h subfamilies which leads to a representation in terms of a Cox point process of genealogies where "points" correspond to single ancestor subfamilies. We determine the Í-valued process conditioned to survive until a finite time T correcting a result from the Ê +-valued literature in the computation of the diffusion coefficient. This is the key ingredient of the excursion law of the Í-valued Feller diffusion. Next we study asymptotics of the Í-valued Feller diffusion conditioned to survive forever and obtain its Kolmogorov-Yaglom limit and show that the limiting processes solve well-posed Í-valued martingale problems. Using infinite divisibility and skew martingale problems we obtain various representations of the long time limits: Í-valued backbone construction of the Palm distribution, the Í-valued version of the Kallenberg tree, the Í-valued version of Feller's branching diffusion with immigration from an immortal lineà la Evans [Eva93]. On the level of Í-valued processes we still have equality (in law) of the Q-process, i.e., the process conditioned to survive up to time T in the limit T → ∞, the size-biased process and Evans' branching process with immigration from an immortal line. The Í-valued generalized quasi-equilibrium is a size-biased version of the Kolmogorov-Yaglom limit law. The above results are key tools for analyzing genealogies in spatial branching populations. We construct the genealogy of the interacting Feller diffusion on a countable group (super random walk) and obtain results on a martingale problem characterization, duality, generalized branching property and the long time behavior for this object. As an application we give a two scale analysis of the super random walk genealogy with strongly recurrent migration providing the asymptotic genealogy of clusters via the Í Ê-valued version of the Dawson-Watanabe process. We indicate the situation in other dimensions. Finally we enrich the Í-valued Feller process further, encoding the information on the whole population ever alive before the present time t and describe its evolution. This leads to the so called fossil process and we relate its limit for t → ∞ to the continuum random tree.

Spinal partitions and invariance under re-rooting of continuum random trees

The Annals of Probability, 2009

We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a two-parameter Poisson-Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform re-rooting.

The probability measure corresponding to 2-plane trees

We study the probability measure µ 0 for which the moment sequence is 3n n 1 n+1. We prove that µ 0 is absolutely continuous, find the density function and prove that µ 0 is infinitely divisible with respect to the additive free convolution.

On the Genealogy of a Population of Biparental Individuals

Journal of Theoretical Biology, 2000

If one goes backward in time, the number of ancestors of an individual doubles at each generation. This exponential growth very quickly exceeds the population size, when this size is finite. As a consequence, the ancestors of a given individual cannot be all different and most remote ancestors are repeated many times in any genealogical tree. The statistical properties of these repetitions in genealogical trees of individuals for a panmictic closed population of constant size N can be calculated. We show that the distribution of the repetitions of ancestors reaches a stationary shape after a small number G c ∝ log N of generations in the past, that only about 80% of the ancestral population belongs to the tree (due to coalescence of branches), and that two trees for individuals in the same population become identical after G c generations have elapsed. Our analysis is easy to extend to the case of exponentially growing population. We used the tree of Edward III which can be found at

On the Uniqueness of the Branching Parameter for a Random Cascade Measure

Journal of Statistical Physics, 2000

An independent random cascade measure µ is specified by a random generator (w 1 , ..., w c ), E w i = 1 where c is the branching parameter. It is shown under certain restrictions that, if µ has two generators with a.s. positive components, and the ratio ln c 1 / ln c 2 for their branching parameters is an irrational number, then µ is a Lebesgue measure. In other words, when c is a power of an integer number p and the p is minimal for c, then a cascade measure that has the property of intermittency specifies p uniquely.

Random Trees and General Branching Processes

Random Structures and Algorithms, 2005

Our proof relies on the fact that considering the evolution of the random tree in continuous time, the process may be viewed as a general branching process, this way classical results can be applied.

Statistical Properties of Genealogical Trees

Physical Review Letters, 1999

We analyse the statistical properties of genealogical trees in a neutral model of a closed population with sexual reproduction and non-overlapping generations. By reconstructing the genealogy of an individual from the population evolution, we measure the distribution of ancestors appearing more than once in a given tree. After a transient time, the probability of repetition follows, up to a rescaling, a stationary distribution which we calculate both numerically and analytically. This distribution exhibits a universal shape with a non-trivial power law which can be understood by an exact, though simple, renormalization calculation. Some real data on human genealogy illustrate the problem, which is relevant to the study of the real degree of diversity in closed interbreeding communities.