arXiv subjects

Osvaldo Angtuncio Hernández

Publications and source records attributed to Osvaldo Angtuncio Hernández.

8 recordsLinked to original sources

arXiv2026

Sampling schemes of multitype continuous-time Bienaymé-Galton-Watson trees and limiting critical genealogies

We study the genealogies of samples of $k$ distinguished particles drawn from the population alive at some fixed time in a continuous-time multitype Bienaymé-Galton-Watson (MBGW) process under two different type dependent sampling schemes: uniform sampling without replacement within types given a fixed type configuration, and sampling according to type-dependent weights. These schemes complement the uniform sampling at fixed time $T$ considered in Angtuncio, Pardo, C. Harris (2026a) which did not distinguish between sampled types. Under each scheme for a fixed sampling time $T$, we characterise the associated times of most recent common ancestors, ancestral offspring distributions, and type-dependent ancestral structure of the sample genealogy. In addition, under the assumption that the MBGW process is critical with finite second moments, we show that, conditional on survival of the population, a large time limiting sample genealogy emerges which is robust to the sampling scheme used. We identify this universal genealogy to have the same tree structure as the single-type case in C. Harris, Johnston, Roberts (2020), and we describe its ancestral type behaviour over scaled-times - this essentially being decoupled from the tree structure except at the times of ancestral splitting events.

math.PR↗

arXiv2026

Uniform sampling of multitype continuous-time Bienaymé-Galton-Watson trees

We study the genealogy of a sample of $k$ individuals taken uniformly without replacement from a continuous-time multitype Bienaymé--Galton--Watson process at fixed times. Our results are quite general, requiring only that the process be non-simple and conservative, and that every type has a positive probability to ``eventually lead to'' all other types within the population. The corresponding single-type case has recently been studied by Johnston (2019), Harris, Johnston, and Roberts (2020), and Harris, Johnston, and Pardo (2024). Our approach is based on a $k$-spine decomposition and a suitable change of measure under which the distinguished spines form a uniform sample at time $T$, while the population size is subject to $k$-size biasing and exponential discounting. This construction preserves a branching Markov property and yields an explicit description of the genealogical tree at fixed times. In particular, we characterise spine splitting times, offspring distributions, and type-dependent ancestral structures, revealing rich interactions between types that are absent in the single-type setting. The present results form the basis of a forthcoming series of papers in which limiting genealogical behaviour is analysed under various asymptotic regimes and more general sampling schemes by the authors, see Angtuncio et al. (2026b), (2026c) and (2026d).

math.PR↗

arXiv2026

Fast Simulation of Size-Constrained Multitype Bienaymé-Galton-Watson Forests and Applications

The degree sequence $(n_{i,j}(k), 1\leq i,j\leq d, k\geq 0)$ of a multitype forest with $d$ types encodes the number of individuals of type $i$ with $k$ children of type $j$. In this paper, we introduce a simple algorithm to sample a multitype forest uniformly from the set of all forests with a given degree sequence (MFGDS). This generalizes the single-type construction of Broutin and Marckert (2014). To achieve this, we extend the Vervaat transform (1979) to multidimensional discrete exchangeable increment processes. We demonstrate that MFGDS extend multitype Bienaymé--Galton--Watson (MBGW) forests. Specifically, mixing MFGDS laws recovers MBGW forests conditioned on a fixed size for each type (CMBGW). Under general assumptions, we derive the law of the total population by types in an MBGW forest and relate it to a multidimensional first-hitting time. This result, which is of independent interest, generalizes the Otter--Dwass (1949,1969) and Kemperman (1950) formulas. By combining this relation with our MFGDS construction, we provide an efficient algorithm to simulate CMBGW forests, generalizing the work of Devroye (2012). When the variance is finite, the expected simulation time outperforms standard naïve methods. For the proof we derive a generalized local limit theorem for multidimensional first-hitting times. Finally, we apply our results to enumerate plane, labeled, and binary multitype forests with fixed sizes, generalizing results of Pitman (1998).

math.PR↗

arXiv2026

Multitype Lévy trees as scaling limits of multitype Bienaymé-Galton-Watson trees

We establish sufficient mild conditions for a sequence of multitype Bienaymé-Galton-Watson trees, conditioned in some sense to be large, to converge to a limiting compact metric space which we call a \emph{multitype Lévy tree}. More precisely, we condition on the size of the maximal subtree of vertices of the same type joined by the root to be large. While we employ a different conditioning, our result can be seen as a generalization to the multitype setting of the continuum random trees defined by Aldous, Duquesne and Le Gall in [Ald91a,Ald91b,Ald93,DLG02]. Our main result is an invariance principle for the convergence of such trees, by gluing single-type Lévy trees together in a method determined by the limiting spectrally positive additive Lévy field, as constructed by Chaumont and Marolleau [CM21]. Our approach is an improvement of a result about the convergence in the Gromov-Hausdorff-Prohorov topology, of compact marked metric spaces equipped with vector-valued measures, which are then glued via an iterative operation. To analyze the gluing operation, we extend the techniques developed by Sénizergues [Sen19,Sen22] to the multitype setting. While the single-type case exhibits a more homogeneous structure with simpler dependency patterns, the multitype case introduces interactions between different types, leading to a more intricate dependency structure where functionals must account for type-specific behaviors and inter-type relationships.

math.PR↗

arXiv2026

Matrix Representations for Scale Functions of Spectrally Negative Lévy Processes with Rational Jumps

For a spectrally negative Lévy process with Laplace transform $ψ$, the $q$-scale function is characterized as the function whose Laplace transform is $(ψ(\cdot)-q)^{-1}$. It has applications in fluctuation theory, for example, exit problems and first hitting probabilities. It is also used in areas like ruin theory, risk theory, continuous state branching processes and optimal control. In this paper, we extend the scale function representation of Ivanovs (2021) from spectrally negative Lévy processes with phase-type jumps to the general case of matrix-exponential jumps. The extension is non-trivial because the probabilistic arguments employed by Ivanovs rely on an embedding to a Markov-modulated Brownian motion, a framework that does not accommodate the algebraic generality of matrix-exponential distributions. We overcome this limitation by embedding the Lévy process into a stochastic fluid process modulated by a rational arrival process (RAP), a class of continuous-valued Markov processes driven by orbit processes. This approach yields iterative schemes related to those of Ivanovs (2021) to provide a simple and explicit formula for the scale function. Our method gives the same fixed point when restricted to the phase-type case, and demonstrates the utility of orbit representations in analytical problems beyond the phase-type setting.

math.PR↗

arXiv2025

Scaling limit of the Aldous-Broder chain on regular graphs: the transient regime

The continuum random tree is the scaling limit of the uniform spanning tree on the complete graph with $N$ vertices. The Aldous-Broder chain on a graph $G=(V,E)$ is a discrete-time stochastic process with values in the space of rooted trees whose vertex set is a subset of $V$ which is stationary under the uniform distribution on the space of rooted trees spanning $G$. In Evans, Pitman and Winter (2006) the so-called root growth with re-grafting process (RGRG) was constructed. Further it was shown that the suitable rescaled Aldous-Broder chain converges to the RGRG weakly with respect to the Gromov-Hausdorff topology. It was shown in Peres and Revelle (2005) that (up to a dimension depending constant factor) the continuum random tree is, with respect to the Gromov-weak topology, the scaling limit of the uniform spanning tree on $\mathbb{Z}_N^d$, $d\ge 5$. This result was recently strengthens in Archer, Nachmias and Shalev (2024) to convergence with respect to the Gromov-Hausdorff-weak topology, and therefore also with respect to the Gromov-Hausdorff topology. In the present paper we show that also the suitable rescaled Aldous-Broder chain converges to the RGRG weakly with respect to the Gromov-Hausdorff topology when initially started in the trivial rooted tree. We give conditions on the increasing graph sequence under which the result extends to regular graphs and give probabilistic expressions scales at which time has to be speed up and edge lengths have to be scaled down.

math.PR↗

arXiv2020

Multidimensional random walks conditioned to stay ordered via generalized ladder height functions

Random walks conditioned to stay positive are a prominent topic in fluctuation theory. One way to construct them is as a random walk conditioned to stay positive up to time $n$, and let $n$ tend to infinity. A second method is conditioning instead to stay positive up to an independent geometric time, and send its parameter to zero. The multidimensional case (condition the components of a $d$-dimensional random walk to be ordered) was solved in [EK08] using the first approach, but some moment conditions need to be imposed. Our approach is based on the second method, which has the advantage to require a minimal restriction, needed only for the finiteness of the $h$-transform in certain cases. We also characterize when the limit is Markovian or sub-Markovian, and give several reexpresions of the $h$-function. Under some conditions given in [Ign18], it can be proved that our $h$-function is the only harmonic function which is zero outside the Weyl chamber $\{x=(x_1,\ldots, x_d)\in \mathbb{R}^d: x_1<\cdots < x_d\}$.

math.PR↗

arXiv2019

Dini derivatives for Exchangeable Increment processes and applications

Let $X$ be an exchangeable increment (EI) process whose sample paths are of infinite variation. We prove that, for any fixed $t$ almost surely, \[ \limsup_{h\to 0 \pm} (X_{t+h}-X_t)/h=\infty \quad\text{and}\quad \liminf_{h\to 0\pm} (X_{t+h}-X_t)/h=-\infty. \]This extends a celebrated result of Rogozin (1968) for Lévy processes, and completes the known picture for finite-variation EI processes. Applications are numerous. For example, we deduce that both half-lines $(-\infty, 0)$ and $(0,\infty)$ are visited immediately for infinite variation EI processes (called upward and downward regularity). We also generalize the zero-one law of Millar (1977) for Lévy processes by showing continuity of $X$ when it reaches its minimum in the infinite variation EI case; an analogous result for all EI processes links right and left continuity at the minimum with upward and downward regularity. We also consider results of Durrett, Iglehart and Miller (1977) on the weak convergence of conditioned Brownian bridges to the normalized Brownian excursion, and broadened to a subclass of Lévy processes and EI processes by Uribe Bravo (2014) and Chaumont and Uribe Bravo (2015). We prove it here for all infinite variation EI processes. We furthermore obtain a description of the convex minorant for non-piecewise linear EI processes, the case of Lévy processes given by Pitman and Uribe Bravo (2012). Our main tool to study the Dini derivatives is a change of measure for EI processes which extends the Esscher transform for Lévy processes.

math.PR↗