arXiv subjects

Jorge González Cázares

Publications and source records attributed to Jorge González Cázares.

13 recordsLinked to original sources

arXiv2026

Infinite-variation Lévy processes are either strongly eroded or abrupt

We prove that every real Lévy process of infinite variation is either abrupt or strongly eroded, resolving the conjectured geometric dichotomy on the smoothness of the convex hull of the graph of a Lévy path [Ann. Inst. H. Poincaré Probab. Stat. 60(4) (2024), 2352-2381]. We give counterexamples to the stronger point-hitting conjecture of Vigon. As applications, we classify convex-hull smoothness, Lipschitz-minorant contact sets and Burgers shock geometry.

math.PR↗

arXiv2026

Robust Scale Estimation in Additive Noise via Weighted Order Statistics

This manuscript develops a non-parametric and robust framework for estimating the scale of additive noise in weakly sparse systems. The method does not require independence, prescribed dependence, or temporal regularity of the noise sequence. We introduce a class of order-statistic estimators based on comparing the sorted observations with deterministic or random proxies generated from a reference noise distribution. This purely spatial approach avoids preliminary filtering or temporal decorrelation, and therefore preserves the sparsity structure of the latent signal. We establish non-asymptotic concentration inequalities for weighted loss functions, with bounds that separate the contribution of the signal from the discrepancy between the ordered noise and the proxy. We then control this proxy discrepancy in independent and correlated regimes, including heavy-tailed reference laws. Finally, we apply the method to high-frequency observations of continuous-time stochastic processes, obtaining scale estimators for fractional Brownian motion and stable Lévy noise in the presence of lower-variation additive perturbations.

math.ST↗

arXiv2026

Understanding the Hamiltonian Monte Carlo through its Physics Fundamentals and Examples

The Hamiltonian Monte Carlo (HMC) algorithm is a powerful Markov Chain Monte Carlo (MCMC) method that uses Hamiltonian dynamics to generate samples from a target distribution. To fully exploit its potential, we must understand how Hamiltonian dynamics work and why they can be used in a MCMC algorithm. This work elucidates the Monte Carlo Hamiltonian, providing comprehensive explanations of the underlying physical concepts. It is intended for readers with a solid foundation in mathematics who may lack familiarity with specific physical concepts, such as those related to Hamiltonian dynamics. Additionally, we provide Python code for the HMC algorithm, examples and comparisons with the Random Walk Metropolis-Hastings (RWMH) algorithm, alongside an exploration of modern variants such as the No-U-Turn Sampler (NUTS), the Dual Averaging (DA) scheme and the repelling-attracting HMC (raHMC), to highlight HMC's strengths and weaknesses when applied to Bayesian Inference.

stat.CO↗

arXiv2025

Multivariate CLT for Lévy processes: convergence rates without moment assumptions

We prove that the norm of a $d$-dimensional Lévy process possesses a finite second moment if and only if the convex distance between an appropriately rescaled process at time $t$ and a standard Gaussian vector is integrable in time with respect to the scale-invariant measure $t^{-1} dt$ on $[1,\infty)$. We further prove that under the standard $\sqrt{t}$-scaling, the corresponding convex distance is integrable if and only if the norm of the Lévy process has a finite $(2+\log)$-moment. Both equivalences also hold for the integrability with respect to $t^{-1} dt$ of the multivariate Kolmogorov distance. Our results imply: (I) polynomial Berry-Esseen bounds on the rate of convergence in the convex distance in the CLT for Lévy processes cannot hold without finiteness of $(2+δ)$-moments for some $δ>0$ and (II) integrability of the convex distance with respect to $t^{-1} dt$ in the domain of non-normal attraction cannot occur for any scaling function.

math.PR↗

arXiv2025

Rates of convergence for multivariate SDEs driven by Lévy processes in the small-time stable domain of attraction

This paper explores the rates of convergence of solutions for multivariate stochastic differential equations (SDEs) driven by Lévy processes within the small-time stable domain of attraction (DoA). Explicit bounds are derived for the uniform Wasserstein distance between solutions of two Lévy-driven SDEs, expressed in terms of driver characteristics. These bounds establish convergence rates in probability for drivers in the DoA, and yield uniform Wasserstein distance convergence for SDEs with additive noise. The methodology uses two couplings for Lévy driver jump components, leading to sharp convergence rates tied to the processes' intrinsic properties.

math.PR↗

arXiv2025

Asymptotically optimal Wasserstein couplings for the small-time stable domain of attraction

We develop two novel couplings between general pure-jump Lévy processes in $\R^d$ and apply them to obtain upper bounds on the rate of convergence in an appropriate Wasserstein distance on the path space for a wide class of Lévy processes attracted to a multidimensional stable process in the small-time regime. We also establish general lower bounds based on certain universal properties of slowly varying functions and the relationship between the Wasserstein and Toscani--Fourier distances of the marginals. Our upper and lower bounds typically have matching rates. In particular, the rate of convergence is polynomial for the domain of normal attraction and slower than a slowly varying function for the domain of non-normal attraction.

math.PR↗

arXiv2025

Bayesian non-parametric survival estimation: stochastic hyperparameter sequences and distribution splicing

A Bayesian non-parametric framework for studying time-to-event data is proposed, where the prior distribution is allowed to depend on an additional random source, and may update with the sample size. Such scenarios are natural, for instance, when considering empirical Bayes techniques or dynamic expert information. In this context, a natural stochastic class for studying the cumulative hazard function are conditionally inhomogeneous independent increment processes with non-decreasing sample paths, also known as mixed time-inhomogeneous subordinators or mixed non-decreasing additive processes. The asymptotic behaviour is studied by showing that Bayesian consistency and Bernstein--von~Mises theorems may be recovered under suitable conditions on the asymptotic negligibility of the stochastic prior sequences. The non-asymptotic behaviour of the posterior is also considered. Namely, upon conditioning, an efficient and exact simulation algorithm for the paths of the Beta Lévy process is provided. As a natural application, it is shown how the model can provide an appropriate definition of non-parametric spliced models. Spliced models target data where an accurate global description of both the body and tail of the distribution is desirable. The Bayesian non-parametric nature of the proposed estimators can offer conceptual and numerical alternatives to their parametric counterparts.

stat.ME↗

arXiv2023

Hölder continuity of the convex minorant of a Lévy process

We characterise the Hölder continuity of the convex minorant of most Lévy processes. The proof is based on a novel connection between the path properties of the Lévy process at zero and the boundedness of the set of $r$-slopes of the convex minorant.

math.PR↗

arXiv2022

Joint density of a stable process and its supremum: regularity and upper bounds

This article uses a combination of three ideas from simulation to establish a nearly optimal polynomial upper bound for the joint density of the stable process and its associated supremum at a fixed time on the entire support of the joint law. The representation of the concave majorant of the stable process and the Chambers-Mallows-Stuck representation for stable laws are used to define an approximation of the random vector of interest. An interpolation technique using multilevel Monte Carlo is applied to accelerate the approximation, allowing us to establish the infinite differentiability of the joint density as well as nearly optimal polynomial upper bounds for the joint mixed derivatives of any order.

math.PR↗

arXiv2022

How smooth can the convex hull of a Lévy path be?

We describe the rate of growth of the derivative $C'$ of the convex minorant of a Lévy path at times where $C'$ increases continuously. Since the convex minorant is piecewise linear, $C'$ may exhibit such behaviour either at the vertex time $τ_s$ of finite slope $s=C'_{τ_s}$ or at time $0$ where the slope is $-\infty$. While the convex hull depends on the entire path, we show that the local fluctuations of the derivative $C'$ depend only on the fine structure of the small jumps of the Lévy process and are the same for all time horizons. In the domain of attraction of a stable process, we establish sharp results essentially characterising the modulus of continuity of $C'$ up to sub-logarithmic factors. As a corollary we obtain novel results for the growth rate at $0$ of meanders in a wide class of Lévy processes.

math.PR↗

arXiv2021

Implementable coupling of Lévy process and Brownian motion

We provide a simple algorithm for construction of Brownian paths approximating those of a Lévy process on a finite time interval. It requires knowledge of the Lévy process trajectory on a chosen regular grid and the law of its endpoint, or the ability to simulate from that. This algorithm is based on reordering of Brownian increments, and it can be applied in a recursive manner. We establish an upper bound on the mean squared maximal distance between the paths and determine a suitable mesh size in various asymptotic regimes. The analysis proceeds by reduction to the comonotonic coupling of increments. Applications to model risk and multilevel Monte Carlo are discussed in detail, and numerical examples are provided.

math.PR↗

arXiv2021

Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation

We develop a computational method for expected functionals of the drawdown and its duration in exponential Lévy models. It is based on a novel simulation algorithm for the joint law of the state, supremum and time the supremum is attained of the Gaussian approximation of a general Lévy process. We bound the bias for various locally Lipschitz and discontinuous payoffs arising in applications and analyse the computational complexities of the corresponding Monte Carlo and multilevel Monte Carlo estimators. Monte Carlo methods for Lévy processes (using Gaussian approximation) have been analysed for Lipschitz payoffs, in which case the computational complexity of our algorithm is up to two orders of magnitude smaller when the jump activity is high. At the core of our approach are bounds on certain Wasserstein distances, obtained via the novel SBG coupling between a Lévy process and its Gaussian approximation. Numerical performance, based on the implementation in the dedicated GitHub repository, exhibits a good agreement with our theoretical bounds.

math.PR↗

arXiv2020

Recovering Brownian and jump parts from high-frequency observations of a Lévy process

We introduce two general non-parametric methods for recovering paths of the Brownian and jump components from high-frequency observations of a Lévy process. The first procedure relies on reordering of independently sampled normal increments and thus avoids tuning parameters. The functionality of this method is a consequence of the small time predominance of the Brownian component, the presence of exchangeable structures, and fast convergence of normal empirical quantile functions. The second procedure amounts to filtering the increments and compensating with the final value. It requires a carefully chosen threshold, in which case both methods yield the same rate of convergence. This rate depends on the small-jump activity and is given in terms of the Blumenthal-Getoor index. Finally, we discuss possible extensions, including the multidimensional case, and provide numerical illustrations.

math.PR↗