arXiv subjects

Gerónimo Uribe Bravo

Publications and source records attributed to Gerónimo Uribe Bravo.

At least 19 recordsLinked to original sources

arXiv2025

Critical branching processes with immigration: scaling limits of local extinction sets

We establish the joint scaling limit of a critical Bienaymé-Galton-Watson process with immigration (BGWI) and its (counting) local time at zero to the corresponding self-similar continuous-state branching process with immigration (CBI) and its (Markovian) local time at zero for balanced offspring and immigration laws in stable domains of attraction. Using a general framework for invariance principles of local times~\cite{MR4463082}, the problem reduces to the analysis of the structure of excursions from zero and positive levels, together with the weak convergence of the hitting times of points of the BGWI to those of the CBI. A key step in the proof of our main limit theorem is a novel Yaglom limit for the law at time $t$ of an excursion with lifetime exceeding $t$ of a scaled infinite-variance critical BGWI. Our main result implies a joint septuple scaling limit of BGWI $Z_1$, its local time at $0$, the random walks $X_1$ and $Y_1$ associated to the reproduction and immigration mechanisms, respectively, the counting local time at $0$ of $X_1$, an additive functional of $Z_1$ and $X_1$ evaluated at this functional. In the septuple limit, four different scaling sequences are identified and given explicitly in terms of the offspring generating function (modulo asymptotic inversion), the local extinction probabilities of the BGWI and the tails of return times to zero of $X_1$.

math.PR↗

arXiv2023

The Sticky Lévy Process as a solution to a Time Change Equation

Stochastic Differential Equations (SDEs) were originally devised by Itô to provide a pathwise construction of diffusion processes. A less explored approach to represent them is through Time Change Equations (TCEs) as put forth by Doeblin. TCEs are a generalization of Ordinary Differential Equations driven by random functions. We present a simple example where TCEs have some advantage over SDEs. We represent sticky Lévy processes as the unique solution to a TCE driven by a Lévy process with no negative jumps. The solution is adapted to the time-changed filtration of the Lévy process driving the equation. This is in contrast to the SDE describing sticky Brownian motion, which is known to have no adapted solutions as first proved by Chitashvili. A known consequence of such non-adaptability for SDEs is that certain natural approximations to the solution of the corresponding SDE do not converge in probability, even though they do converge weakly. Instead, we provide strong approximation schemes for the solution of our TCE (by adapting Euler's method for ODEs), whenever the driving Lévy process is strongly approximated.

math.PR↗

arXiv2022

A Meyer-Itô Formula for Stable Processes via Fractional Calculus

The infinitesimal generator of a one-dimensional strictly $α$-stable process can be represented as a weighted sum of (right and left) Riemann-Liouville fractional derivatives of order $α$ and one obtains the fractional Laplacian in the case of symmetric stable processes. Using this relationship, we compute the inverse of the infinitesimal generator on Lizorkin space, from which we can recover the potential if $α\in (0,1)$ and the recurrent potential if $α\in (1,2)$. The inverse of the infinitesimal generator is expressed in terms of a linear combination of (right and left) Riemann-Liouville fractional integrals of order $α$. One can then state a class of functions that give semimartingales when applied to strictly stable processes and state a Meyer-Itô theorem with a non-zero (occupational) local time term, providing a generalization of the Tanaka formula given by Tsukada (2019). This result is used to find a Doob-Meyer (or semimartingale) decomposition for $|X_t - x|^γ$ with $X$ a recurrent strictly stable process of index $α$ and $γ\in (α-1,α)$, generalizing the work of Engelbert and Kurenok (2019) to the asymmetric case.

math.PR↗

arXiv2021

Geometrically Convergent Simulation of the Extrema of Lévy Processes

We develop a novel approximate simulation algorithm for the joint law of the position, the running supremum and the time of the supremum of a general Lévy process at an arbitrary finite time. We identify the law of the error in simple terms. We prove that the error decays geometrically in $L^p$ (for any $p\geq 1$) as a function of the computational cost, in contrast with the polynomial decay for the approximations available in the literature. We establish a central limit theorem and construct non-asymptotic and asymptotic confidence intervals for the corresponding Monte Carlo estimator. We prove that the multilevel Monte Carlo estimator has optimal computational complexity (i.e. of order $ε^{-2}$ if the mean squared error is at most $ε^2$) for locally Lipschitz and barrier-type functionals of the triplet and develop an unbiased version of the estimator. We illustrate the performance of the algorithm with numerical examples.

math.PR↗

arXiv2020

An algorithm for simulating Brownian increments on a sphere

This paper presents a novel formula for the transition density of the Brownian motion on a sphere of any dimension and discusses an algorithm for the simulation of the increments of the spherical Brownian motion based on this formula. The formula for the density is derived from an observation that a suitably transformed radial process (with respect to the geodesic distance) can be identified as a Wright-Fisher diffusion process. Such processes satisfy a duality (a kind of symmetry) with a certain coalescent processes and this in turn yields a spectral representation of the transition density, which can be used for exact simulation of their increments using the results of Jenkins and Spanò (2017). The symmetry then yields the algorithm for the simulation of the increments of the Brownian motion on a sphere. We analyse the algorithm numerically and show that it remains stable when the time-step parameter is not too small.

cond-mat.stat-mech↗

arXiv2020

On the profile of trees with a given degree sequence

A degree sequence is a sequence ${\bf s}=(N_i,i\geq 0)$ of non-negative integers satisfying $1+\sum_i iN_i=\sum_i N_i<\infty$. We are interested in the uniform distribution $\mathbb{P}_{\bf s}$ on rooted plane trees whose degree sequence equals ${\bf s}$, giving conditions for the convergence of the profile (sequence of generation sizes) as the size of the tree goes to infinity. This provides a more general formulation and a probabilistic proof of a conjecture due to Aldous (1991). Our formulation contains and extends results in this direction obtained previously by Drmota and Gittenberger (1997) and Kersting (2011). A technical result is needed to ensure that trees with law $\mathbb{P}_{\bf s}$ have enough individuals in the first generations, and this is handled through novel path transformations and fluctuation theory of exchangeable increment processes. As a consequence, we obtain a boundedness criterion for the inhomogeneous continuum random tree introduced by Aldous, Miermont and Pitman (2004).

math.PR↗

arXiv2019

$\varepsilon$-strong simulation of the convex minorants of stable processes and meanders

Using marked Dirichlet processes we characterise the law of the convex minorant of the meander for a certain class of Lévy processes, which includes subordinated stable and symmetric Lévy processes. We apply this characterisaiton to construct $\varepsilon$-strong simulation ($\varepsilon$SS) algorithms for the convex minorant of stable meanders, the finite dimensional distributions of stable meanders and the convex minorants of weakly stable processes. We prove that the running times of our $\varepsilon$SS algorithms have finite exponential moments. We implement the algorithms in Julia 1.0 (available on GitHub) and present numerical examples supporting our convergence results.

math.PR↗

arXiv2019

Invariance principles for local times in regenerative settings

Consider a stochastic process $\mathfrak{X}$, regenerative at a state $x$ which is instantaneous and regular. Let $L$ be a regenerative local time for $\mathfrak{X}$ at $x$. Suppose furthermore that $\mathfrak{X}$ can be approximated by discrete time regenerative processes $\mathfrak{X}^n$ for which $x$ is accesible. We give conditions on $\mathfrak{X}$ and $\mathfrak{X}^n$ so that the naturally defined local time of $\mathfrak{X}^n$ converges weakly to $L$. This limit theorem generalizes previous invariance principles that have appeared in the literature.

math.PR↗

arXiv2019

A note on the exact simulation of spherical Brownian motion

We describe an exact simulation algorithm for the increments of Brownian motion on a sphere of arbitrary dimension, based on the skew-product decomposition of the process with respect to the standard geodesic distance. The radial process is closely related to a Wright-Fisher diffusion, increments of which can be simulated exactly using the recent work of Jenkins & Spanò (2017) [JS17]. The rapid spinning phenomenon of the skew-product decomposition then yields the algorithm for the increments of the process on the sphere.

math.PR↗

arXiv2019

Exact Simulation of the Extrema of Stable Processes

We exhibit an exact simulation algorithm for the supremum of a stable process over a finite time interval using dominated coupling from the past (DCFTP). We establish a novel perpetuity equation for the supremum (via the representation of the concave majorants of Lévy processes) and apply it to construct a Markov chain in the DCFTP algorithm. We prove that the number of steps taken backwards in time before the coalescence is detected is finite. We analyse numerically the performance of the algorithm (the code, written in Julia 1.0, is available on GitHub).

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↗

arXiv2019

Random walks with preferential relocations and fading memory: a study through random recursive trees

Consider a stochastic process that behaves as a $d$-dimensional simple and symmetric random walk, except that, with a certain fixed probability, at each step, it chooses instead to jump to a given site with probability proportional to the time it has already spent there. This process has been analyzed in the physics literature under the name "random walk with preferential relocations", where it is argued that the position of the walker after $n$ steps, scaled by $\log n$, converges to a Gaussian random variable; because of the $\log$ spatial scaling, the process is said to undergo a "slow diffusion". In this paper, we generalize this model by allowing the underlying random walk to be any Markov process and the random run-lengths (time between two relocations) to be i.i.d.-distributed. We also allow the memory of the walker to fade with time, meaning that when a relocations occurs, the walker is more likely to go back to a place it has visited more recently. We prove rigorously the central limit theorem described above (plus a local limit theorem and the convergence of the weighted occupation measure) by associating to the process a growing family of vertex-weighted random recursive trees and a Markov chain indexed by this tree. The spatial scaling of our relocated random walk is related to the height of a "typical" vertex in the random tree. This typical height can range from doubly-logarithmic to logarithmic or even a power of the number of nodes of the tree, depending on the form of the memory.

math.PR↗

arXiv2018

Projections of spherical Brownian motion

We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R}^{n+\ell}$. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of existence and pathwise uniqueness and show that they always hold. The square of the radial component is a Wright-Fisher diffusion with mutation and it features in a skew-product decomposition of the projected spherical Brownian motion. A more general SDE on the unit ball in $\mathbb{R}^{n+\ell}$ allows us to geometrically realize the Wright-Fisher diffusion with general non-negative parameters as the radial component of its solution.

math.PR↗

arXiv2018

Totally Ordered Measured Trees and Splitting Trees with Infinite Variation II: Prolific Skeleton Decomposition

The first part of this paper ( arXiv:1607.02114 ) introduced splitting trees, those chronological trees admitting the self-similarity property where individuals give birth, at constant rate, to iid copies of themselves. It also established the intimate relationship between splitting trees and Lévy processes. The chronological trees involved were formalized as Totally Ordered Measured (TOM) trees. The aim of this paper is to continue this line of research in two directions: we first decompose locally compact TOM trees in terms of their prolific skeleton (consisting of its infinite lines of descent). When applied to splitting trees, this implies the construction of the supercritical ones (which are locally compact) in terms of the subcritical ones (which are compact) grafted onto a Yule tree (which corresponds to the prolific skeleton). As a second (related) direction, we study the genealogical tree associated to our chronological construction. This is done through the technology of the height process introduced by Duquesne and Le Gall. In particular we prove a Ray-Knight type theorem which extends the one for (sub)critical Lévy trees to the supercritical case.

math.PR↗

arXiv2017

Totally Ordered Measured Trees and Splitting Trees with Infinite Variation

Combinatorial trees can be used to represent genealogies of asexual individuals. These individuals can be endowed with birth and death times, to obtain a so-called `chronological tree'. In this work, we are interested in the continuum analogue of chronological trees in the setting of real trees. This leads us to consider totally ordered and measured trees, abbreviated as TOM trees. First, we define an adequate space of TOM trees and prove that under some mild conditions, every compact TOM tree can be represented in a unique way by a so-called contour function, which is right-continuous, admits limits from the left and has non-negative jumps. The appropriate notion of contour function is also studied in the case of locally compact TOM trees. Then we study the splitting property of (measures on) TOM trees which extends the notion of `splitting tree' studied in \cite{MR2599603}, where during her lifetime, each individual gives birth at constant rate to independent and identically distributed copies of herself. We prove that the contour function of a TOM tree satisfying the splitting property is associated to a spectrally positive Lévy process that is not a subordinator, both in the critical and subcritical cases of compact trees as well as in the supercritical case of locally compact trees.

math.PR↗

arXiv2015

Affine processes on $\mathbb{R}_+^n \times \mathbb{R}^n$ and multiparameter time changes

We present a time change construction of affine processes with state-space $\mathbb{R}_+^m\times \mathbb{R}^n$. These processes were systematically studied in (Duffie, Filipović and Schachermayer, 2003) since they contain interesting classes of processes such as Lévy processes, continuous branching processes with immigration, and of the Ornstein-Uhlenbeck type. The construction is based on a (basically) continuous functional of a multidimensional Lévy process which implies that limit theorems for Lévy processes (both almost sure and in distribution) can be inherited to affine processes. The construction can be interpreted as a multiparameter time change scheme or as a (random) ordinary differential equation driven by discontinuous functions. In particular, we propose approximation schemes for affine processes based on the Euler method for solving the associated discontinuous ODEs, which are shown to converge.

math.PR↗

arXiv2014

Local extinction in continuous-state branching processes with immigration

The purpose of this article is to observe that the zero sets of continuous-state branching processes with immigration (CBI) are infinitely divisible regenerative sets. Indeed, they can be constructed by the procedure of random cutouts introduced by Mandelbrot in 1972. We then show how very precise information about the zero sets of CBI can be obtained in terms of the branching and immigrating mechanism.

math.PR↗

arXiv2014

Shifting processes with cyclically exchangeable increments at random

We propose a path transformation which applied to a cyclically exchangeable increment process conditions its minimum to belong to a given interval. This path transformation is then applied to processes with start and end at zero. It is seen that, under simple conditions, the weak limit as epsilon tends to zero of the process conditioned on remaining above minus epsilon exists and has the law of the Vervaat transformation of the process. We examine the consequences of this path transformation on processes with exchangeable increments, Lévy bridges, and the Brownian bridge.

math.PR↗