arXiv subjects

Sandra Palau

Publications and source records attributed to Sandra Palau.

At least 19 recordsLinked to original sources

arXiv2025

Distributional properties of first jump times of CBI processes with jump sizes in given Borel sets

We derive an expression for the joint distribution function of the first jump times of a continuous state and continuous time branching process with immigration (CBI process) with jump sizes in given Borel sets having finite total L\'evy measures, which is defined as the sum of the measures appearing in the branching and immigration mechanisms of the CBI process in question. Our result generalizes a corresponding result of He and Li (2016), who considered this problem in case of a single Borel set having finite total L\'evy measure.

math.PR↗

arXiv2025

The coalescent of a sample from a linear-fractional branching process

In this article, we focus on Bienaym\'e-Galton-Watson processes with linear-fractional offspring distributions. At a fixed generation, we consider a sample of the individuals alive, drawn in two different ways: either through Bernoulli sampling, where each individual is selected independently with a given probability, or through uniform sampling, where a fixed number of individuals are chosen uniformly at random. We analyze the genealogical trees generated by the sampled individuals. In particular, we establish a relationship between the distributions of the trees resulting from the two sampling schemes.

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arXiv2024

The largest fragment in self-similar fragmentation processes of positive index

We study a self-similar fragmentation process with dislocation measure $\nu$ and self-similarity index $\alpha > 0$. Let $e^{-m_t}$ denote the size of the largest fragment at time $t \geq 0$. For dislocation measures satisfying a regularity condition of the form $\nu(1 - s_1 > \delta) = \delta^{-\theta} \ell(1/\delta)$ with $\theta \in [0,1)$ and slowly varying $\ell$, we prove almost sure convergence \[ \lim_{t \to \infty} (m_t - g(t)) = 0, \] where $g(t) = (\log t - (1 - \theta) \log \log t + f(t))/\alpha$, and $f(t) = o(\log \log t)$ is a lower order correction that can be described explicitly in terms of $\ell$ and $\theta$. Our results sharpen substantially the best prior result on general self-similar fragmentation processes, due to Bertoin, which states that $m_t = (1+o(1)) \log (t)/\alpha$.

math.PR↗

arXiv2023

Fixation and stationary times for the $\Lambda$-Wright-Fisher process

We study the fixation and stationary behavior of the Lambda-Wright-Fisher process with parent-independent mutation and finitely many types, a jump-diffusion model for allele frequency dynamics in large populations with potentially large offspring variance. Using a lookdown construction, we characterize the distribution of fixation times and the order of allele extinctions in the absence of mutations, and identify a strong stationary time in the presence of mutations. Our results include explicit expressions for the mean fixation and stationary times for the Wright-Fisher diffusion, and mean fixation times in the Beta-coalescent case. A key component of our approach is the analysis of the fixation line introduced by H\'enard in (Ann. Appl. Probab., 25:3007-3032, 2015). We extend this process to incorporate mutation, providing a unified framework for studying both fixation and equilibrium behavior.

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arXiv2023

Distributional properties of jumps of multi-type CBI processes

We study the distributional properties of jumps of multi-type continuous state and continuous time branching processes with immigration (multi-type CBI processes). We derive an expression for the distribution function of the first jump time of a multi-type CBI process with jump size in a given Borel set having finite total L\'evy measure, which is defined as the sum of the measures appearing in the branching and immigration mechanisms of the multi-type CBI process in question. Using this we derive an expression for the distribution function of the local supremum of the norm of the jumps of a multi-type CBI process. Further, we show that if $A$ is a nondegenerate rectangle anchored at zero and with total L\'evy measure zero, then the probability that the local coordinate-wise supremum of jumps of the multi-type CBI process belongs to $A$ is zero. We also prove that a converse statement holds.

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arXiv2023

Rates on Yaglom's limit for Galton-Watson processes in a varying environment

A Galton-Watson process in a varying environment is a discrete time branching process where the offspring distributions vary among generations. It is known that in the critical case, these processes have a Yaglom limit, that is, a suitable normalization of the process conditioned on non-extinction converges in distribution to a standard exponential random variable. In this manuscript, we provide the rate of convergence of the Yaglom limit with respect to the Wasserstein metric.

math.PR↗

arXiv2022

On multitype Branching Processes with Interaction

Motivated by the stochastic Lotka-Volterra model, we introduce discrete-state interacting multitype branching processes. We show that they can be obtained as the sum of a multidimensional random walk with a Lamperti-type change proportional to the population size; and a multidimensional Poisson process with a time-change proportional to the pairwise interactions. We define the analogous continuous-state process as the unique strong solution of a multidimensional SDE. We prove that the scaling limits of the discrete-state process correspond to its continuous counterpart. In addition, we show that the continuous-state model can be constructed as a generalized Lamperti-type transformation of multidimensional Lévy processes.

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arXiv2022

The coalescent structure of Galton-Watson trees in varying environments

We investigate the genealogy of a sample of $k\geq1$ particles chosen uniformly without replacement from a population alive at large times in a critical discrete-time Galton-Watson process in a varying environment (GWVE). We will show that subject to an explicit deterministic time-change involving only the mean and variances of the varying offspring distributions, the sample genealogy always converges to the same universal genealogical structure; it has the same tree topology as Kingman's coalescent, and the coalescent times of the $k-1$ pairwise mergers look like a mixture of independent identically distributed times. Our approach uses $k$ distinguished \emph{spine} particles and a suitable change of measure under which (a) the spines form a uniform sample without replacement, as required, but additionally (b) there is $k$-size biasing and discounting according to the population size. Our work significantly extends the spine techniques developed in Harris, Johnston, and Roberts \emph{[Annals Applied Probability, 2020]} for genealogies of uniform samples of size $k$ in near-critical continuous-time Galton-Watson processes, as well as a two-spine GWVE construction in Cardona and Palau \emph{[Bernoulli, 2021]}. Our results complement recent works by Kersting \emph{[Proc. Steklov Inst. Maths., 2022]} and Boenkost, Foutel-Rodier, and Schertzer \emph{[arXiv:2207.11612]}.

math.PR↗

arXiv2022

Coalescent point process of branching trees in varying environment

Consider an arbitrary large population at the present time, originated at an unspecified arbitrary large time in the past, where individuals in the same generation reproduce independently, forward in time, with the same offspring distribution but potentially changing among generations. In other words, the reproduction is driven by a Galton-Watson process in a varying environment. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process $(A_i, i\geq 1)$, where $A_i$ is the coalescent time between individuals $i$ and $i+1$. In general, this process is not Markov. In constant environment, Lambert and Popovic (2013) proposed a Markov process of point measures to reconstruct the coalescent point process. We present a counterexample where we show that their process does not have the Markov property. The main contribution of this work is to propose a vector valued Markov process $(B_i,i\geq 1)$, that reach the goal to reconstruct the genealogy, with finite information for every $i$. Additionally, when the offspring distributions are lineal fractional, we show that the variables $(A_i, i\geq 1)$ are independent and identically distributed.

math.PR↗

arXiv2021

Attraction to and repulsion from a subset of the unit sphere for isotropic stable Lévy processes

Taking account of recent developments in the representation of $d$-dimensional isotropic stable Lévy processes as self-similar Markov processes, we consider a number of new ways to condition its path. Suppose that $Ω$ is a region of the unit sphere $\mathbb{S}^{d-1} = \{x\in \mathbb{R}^d: |x| =1\}$. We construct the aforesaid stable Lévy process conditioned to approach $\mathsf{S}$ continuously from either inside or outside of the sphere. Additionally, we show that %this these processes are in duality with the stable process conditioned to remain inside the sphere and absorb continuously at the origin and to remain outside of the sphere, respectively. Our results extend the recent contributions of Döring and Weissman (2018),, where similar conditioning is considered, albeit in one dimension. As is the case there, we appeal to recent fluctuation identities related to the deep factorisation of stable processes.

math.PR↗

arXiv2021

Asymptotic behavior of projections of supercritical multi-type continuous state and continuous time branching processes with immigration

Under a fourth order moment condition on the branching and a second order moment condition on the immigration mechanisms, we show that an appropriately scaled projection of a supercritical and irreducible continuous state and continuous time branching process with immigration on certain left non-Perron eigenvectors of the branching mean matrix is asymptotically mixed normal. With an appropriate random scaling, under some conditional probability measure, we prove asymptotic normality as well. In case of a non-trivial process, under a first order moment condition on the immigration mechanism, we also prove the convergence of the relative frequencies of distinct types of individuals on a suitable event; for instance, if the immigration mechanism does not vanish, then this convergence holds almost surely.

math.PR↗

arXiv2020

Oscillatory attraction and repulsion from a subset of the unit sphere or hyperplane for isotropic stable Lévy processes

Suppose that $\mathsf{S}$ is a closed set of the unit sphere $\mathbb{S}^{d-1} = \{x\in \mathbb{R}^d: |x| =1\}$ in dimension $d\geq2$, which has positive surface measure. We construct the law of absorption of an isotropic stable Lévy process in dimension $d\geq2$ conditioned to approach $\mathsf{S}$ continuously, allowing for the interior and exterior of $\mathbb{S}^{d-1}$ to be visited infinitely often. Additionally, we show that this process is in duality with the underlying stable Lévy process. We can replicate the aforementioned results by similar ones in the setting that $\mathsf{S}$ is replaced by $\mathsf{D}$, a closed bounded subset of the hyperplane $\{x\in\mathbb{R}^d : (x, v) = 0\}$ with positive surface measure, where $v$ is the unit orthogonal vector and where $(\cdot,\cdot )$ is the usual Euclidean inner product. Our results complement similar results of the authors Kyprianou, Palau and Saizmaa (2020) in which the stable process was further constrained to attract to and repel from $\mathsf{S}$ from either the exterior or the interior of the unit sphere.

math.PR↗

arXiv2020

Yaglom's limit for critical Galton-Watson processes in varying environment: A probabilistic approach

A Galton-Watson process in varying environment is a discrete time branching process where the offspring distributions vary among generations. Based on a two-spine decomposition technique, we provide a probabilistic argument of a Yaglom-type limit for this family processes. The result states that, in the critical case, a suitable normalisation of the process conditioned on non-extinction converges in distribution to an exponential random variable. Recently, this result has been established by Kersting [{\it J. Appl. Probab.} {\bf57}(1), 196--220, 2020] using analytic techniques.

math.PR↗

arXiv2019

Almost sure, L_1- and L_2-growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration

Under a first order moment condition on the immigration mechanism, we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration (CBI process) converges almost surely. If an $x \log(x)$ moment condition on the branching mechanism does not hold, then the limit is zero. If this $x \log(x)$ moment condition holds, then we prove $L_1$ convergence as well. The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing. If, in addition, a suitable extra power moment condition on the branching mechanism holds, then we provide the correct scaling for the projection of a CBI process on certain left non-Perron eigenvectors of the branching mean matrix in order to have almost sure and $L_1$ limit. Moreover, under a second order moment condition on the branching and immigration mechanisms, we prove $L_2$ convergence of an appropriately scaled process and the above mentioned projections as well. A representation of the limits is also provided under the same moment conditions.

math.PR↗

arXiv2019

Law of large numbers for supercritical superprocesses with non-local branching

In this paper we establish a weak and a strong law of large numbers for supercritical superprocesses with general non-local branching mechanisms. Our results complement earlier results obtained for superprocesses with only local branching. Several interesting examples are developed, including multitype continuous-state branching processes, multitype superdiffusions and superprocesses with discontinuous spatial motions and non-decomposable branching mechanisms.

math.PR↗

arXiv2018

Backbone decomposition of multitype superprocesses

In this paper, we provide a construction of the so-called backbone decomposition for multitype supercritical superprocesses. While backbone decompositions are fairly well-known for both continuous-state branching processes and superprocesses in the one-type case, so far no such decompositions or even description of prolific genealogies have been given for the multitype cases. Here we focus on superprocesses, but by turning the movement off, we get the prolific backbone decomposition for multitype continuous-state branching processes as an easy consequence of our results.

math.PR↗

arXiv2018

Almost sure growth of supercritical multi-type continuous state branching process

In Li (2011), Example 2.2, the notion of a multi-type continuous-state branching process (MCSBP) was introduced with a finite number of types, with the countably infinite case being proposed in Kyprianou and Palau (2017). One may consider such processes as a super-Markov chain on a countable state-space of types, which undertakes both local and non-local branching. In Kyprianou and Palau (2017) it was shown that, for MCSBPs, under mild conditions, there exists a lead eigenvalue which characterises the spectral radius of the linear semigroup associated to the process. Moreover, in a qualitative sense, the sign of this eigenvalue distinguishes between the cases where there is local extinction and exponential growth. In this paper, we continue in this vein and show that, when the number of types is finite, the lead eigenvalue gives the precise almost sure rate of growth of each type. This result matches perfectly classical analogues for multi-type Galton--Watson processes.

math.PR↗

arXiv2018

Extinction properties of multi-type continuous-state branching processes

Recently in Barczy, Li and Pap (2015), the notion of a multi-type continuous-state branching process (with immigration) having d-types was introduced as a solution to an d-dimensional vector- valued SDE. Preceding that, work on affine processes, originally motivated by math- ematical finance, in Duffie, Filipovic and Schachermayer (2003) also showed the existence of such processes. See also more recent contributions in this direction due to Gabrielli and Teichmann (2014) and Caballero, Perez Garmendia and Uribe Bravo (2015). Older work on multi-type continuous-state branching processes is more sparse but includes Watanabe (1969) and Ma (2013), where only two types are considered. In this paper we take a completely different approach and consider multi-type continuous-state branching process, now allowing for up to a countable infinity of types, defined instead as a super Markov chain with both local and non-local branching mechanisms. In the spirit of Englander and Kyprianou (2004) we explore their extinction properties and pose a number of open problems.

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