arXiv subjects

Arturo Jaramillo

Publications and source records attributed to Arturo Jaramillo.

At least 19 recordsLinked to original sources

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

Non-commutative law of rare events

We establish quantitative versions of the law of rare events and binomial approximations in non-commutative probability settings, including the free, Boolean, and monotone convolution frameworks. Our main results provide explicit error bounds in the non-commutative Wasserstein distance for approximations of convolutions of rare countings by non-commutative Poisson and binomial distributions. These bounds extend classical results from the tensor setting to the non-commutative regime. Our approach relies on a discrete Lindeberg-type interpolation scheme combined with algebraic properties of cumulants adapted to each independence notion. The results presented here fill a gap in the literature concerning explicit rates of convergence in non-commutative limit theorems.

math.OA↗

arXiv2026

Adjusted Wasserstein distances for bridging empirical and true distributions with applications to MDS

This paper examines how metric adjustments to Multidimensional Scaling (MDS) can enhance its effectiveness as a visual tool for pattern recognition. The distance under consideration, referred to as Max-D-SW, is an adjustment of the Max-Sliced Wasserstein distance. In contrast to the original formulation, which optimizes over single unit directions, Max-D-SW aggregates contributions over orthonormal bases. This modification provides a clear numerical advantage in MDS outcomes, particularly when applied to heavy-tailed distributions. We also establish sample-complexity bounds showing that Max-D-SW remains statistically tractable, with rates comparable to those of its max-sliced counterpart. Moreover, we show that a better sample complexity for a metric does not necessarily translate into better performance when the metric is used as an input for MDS.

stat.ML↗

arXiv2026

Attenuated Poisson Dirichlet approximations for divisibility configurations

We study the point process formed by the normalized logarithms of the distinct prime factors of a harmonic random sample. We prove a quantitative convergence result, in a Wasserstein-type metric over decreasing sequences, toward the atom sequence of a Dickman Poisson cloud conditioned to have total mass at most one, equivalently a uniformly attenuated Poisson-Dirichlet law. The proof is based on the conditioned geometric representation of harmonic samples, a Poisson approximation chain for the associated point processes, monotone couplings of Poisson point processes, and Kolmogorov estimates for the Dickman approximation of weighted geometric sums.

math.PR↗

arXiv2026

Distributional comparison for non-commutative infinitely divisible probability measures

We determine ``cumulant-type'' upper bounds of the non-commutative Wasserstein distance for certain classes of distributions $μ$ and $ν$, which are infinite divisible with respect to the Boolean, classical and free convolutions. The main contribution of the manuscript is an estimation of the non-commutative Wasserstein distance between $μ$ and $ν$, expressed in terms of the difference between cumulants of order less than $2m+4$.

math.PR↗

arXiv2026

Hockey-Stick Domination and Distributional Comparison on Finite Posets

We develop a framework for comparing probability measures on finite posets via hockey-stick domination, an order relation defined through interval-counting test functions. The theory introduces poset integrals, derivatives, power functions and the associated moment functionals, all of which are invariant under poset isomorphisms. We prove that hockey-stick domination admits an exact quantitative characterization: whenever $μ$ is dominated by $ν$ in the hockey-stick order, the corresponding Zolotarev-type distance is equal to one half of the second-order poset moment of $ν-μ$. We further develop a constructive theory for generating such domination relations. In particular, we show that hockey-stick domination is preserved under direct products, disjoint unions, ordinal sums, and suitable ideal restrictions, yielding natural families of examples on chains, Boolean posets, rectangular lattices, rooted trees, and Young diagrams.

math.CO↗

arXiv2026

Additive functionals of Harmonic samples: the conditioned Dickman regime

We study the distributional behavior of additive arithmetic functions evaluated at integers drawn from the harmonic distribution. Our main result shows that, for a broad family of completely additive functions, their evaluations at harmonic samples, suitably normalized, converge in law to conditioned Dickman-type Poisson integrals. This behavior contrasts with the Gaussian limits arising in the classical Erdös-Kac theorem under uniform sampling. Our approach combines the probabilistic representation of harmonic samples via independent geometric variables, analytic inputs such as Mertens' approximation, and a Poissonization procedure.

math.NT↗

arXiv2025

Asymptotics for additive functionals of particle systems via Stein's method

We consider additive functionals of systems of random measures whose initial configuration is given by a Poisson point process, and whose individual components evolve according to arbitrary Markovian or non-Markovian measure valued dynamics, with no structural assumptions beyond basic moment bounds. In this setting and under adequate conditions, we establish a general third moment theorem for the normalized functionals. Building on this result, we obtain the first quantitative bounds in the Wasserstein distance for a variety of moving-measure models initialized by Poisson-driven clouds of points, turning qualitative central limit theorems into explicit rates of convergence. The scope of the approach is then demonstrated through several examples, including systems driven by fractional Brownian motion, $α$-stable processes, uniformly elliptic diffusions, and spectral empirical measures arising from Dyson Brownian motion, all under broad assumptions on the control measure of the initial Poisson configuration. The analysis relies on a combination of Stein's method with Mecke's formula, in the spirit of the Poisson Malliavin-Stein methodology.

math.PR↗

arXiv2025

Branching Stein Variational Gradient Descent for sampling multimodal distributions

We propose a novel particle-based variational inference method designed to work with multimodal distributions. Our approach, referred to as Branched Stein Variational Gradient Descent (BSVGD), extends the classical Stein Variational Gradient Descent (SVGD) algorithm by incorporating a random branching mechanism that encourages the exploration of the state space. In this work, a theoretical guarantee for the convergence in distribution is presented, as well as numerical experiments to validate the suitability of our algorithm. Performance comparisons between the BSVGD and the SVGD are presented using the Wasserstein distance between samples and the corresponding computational times.

cs.LG↗

arXiv2025

Quantitative and stable limits of high-frequency statistics of Lévy processes: a Stein's method approach

We establish inequalities for assessing the distance between the distribution of errors of partially observed high-frequency statistics of multidimensional Lévy processes and that of a mixed Gaussian random variable. Furthermore, we provide a general result guaranteeing stable functional convergence. Our arguments rely on a suitable adaptation of the Stein's method perspective to the context of mixed Gaussian distributions, specifically tailored to the framework of high-frequency statistics.

math.PR↗

arXiv2025

Approximation of Smooth Numbers for Harmonic Samples A Stein method Approach

We present a de Bruijn type approximation for quantifying the content of m smooth numbers, derived from samples obtained through a probability measure over the set of integers less than or equal to n, with point mass function at k inversely proportional to k. Our analysis is based on a stochastic representation of the measure of interest, utilizing weighted independent geometric random variables. This representation is analyzed through the lens of Stein method for the Dickman distribution. A pivotal element of our arguments relies on precise estimations concerning the regularity properties of the solution to the Dickman Stein equation for heaviside functions, recently developed by Bhattacharjee and Schulte. Remarkably, our arguments remain mostly in the realm of probability theory, with Mertens first and third theorems standing as the only number theory estimations required.

math.PR↗

arXiv2024

Non-commutative Stein's Method: Applications to Free Probability and Sums of Non-commutative Variables

We present a straightforward formulation of Stein's method for the semicircular distribution, specifically designed for the analysis of non-commutative random variables. Our approach employs a non-commutative version of Stein's heuristic, interpolating between the target and approximating distributions via the free Ornstein-Uhlenbeck semigroup. A key application of this work is to provide a new perspective for obtaining precise estimates of accuracy in the semicircular approximation for sums of weakly dependent variables, measured under the total variation metric. We leverage the simplicity of our arguments to achieve robust convergence results, including: (i) A Berry-Esseen theorem under the total variation distance and (ii) Enhancements in rates of decay under the non-commutative Wasserstein distance towards the semicircular distribution, given adequate high-order moment matching conditions.

math.PR↗

arXiv2024

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↗

arXiv2024

Optimal estimation of local time and occupation time measure for an α-stable Levy process

We present a novel theoretical result on estimation of local time and occupation time measure of an α-stable Lévy process with α in (1, 2). Our approach is based upon computing the conditional expectation of the desired quantities given high frequency data, which is an L^2-optimal statistic by construction. We prove the corresponding stable central limit theorems and discuss a statistical application. In particular, this work extends the results of [Ivanovs and i Podolskij (2021)], which investigated the case of the Brownian motion.

math.PR↗

arXiv2022

Quantitative limit theorems via relative log-concavity

In this paper we develop tools for studying limit theorems by means of convexity. We establish bounds for the discrepancy in total variation between probability measures $μ$ and $ν$ such that $ν$ is log-concave with respect to $μ$. We discuss a variety of applications, which include geometric and binomial approximations to sums of random variables, and discrepancy between Gamma distributions. As special cases we obtain a law of rare events for intrinsic volumes, quantitative bounds on proximity to geometric for infinitely divisible distributions, as well as binomial and Poisson approximation for matroids.

math.PR↗

arXiv2022

Fluctuations for matrix-valued Gaussian processes

We consider a symmetric matrix-valued Gaussian process $Y^{(n)}=(Y^{(n)}(t);t\ge0)$ and its empirical spectral measure process $μ^{(n)}=(μ_{t}^{(n)};t\ge0)$. Under some mild conditions on the covariance function of $Y^{(n)}$, we find an explicit expression for the limit distribution of $$Z_F^{(n)} := \left( \big(Z_{f_1}^{(n)}(t),\ldots,Z_{f_r}^{(n)}(t)\big) ; t\ge0\right),$$ where $F=(f_1,\dots, f_r)$, for $r\ge 1$, with each component belonging to a large class of test functions, and $$ Z_{f}^{(n)}(t) := n\int_{\mathbb{R}}f(x)μ_{t}^{(n)}(\text{d} x)-n\mathbb{E}\left[\int_{\mathbb{R}}f(x)μ_{t}^{(n)}(\text{d} x)\right].$$ More precisely, we establish the stable convergence of $Z_F^{(n)}$ and determine its limiting distribution. An upper bound for the total variation distance of the law of $Z_{f}^{(n)}(t)$ to its limiting distribution, for a test function $f$ and $t\geq0$ fixed, is also given.

math.PR↗

arXiv2021

A generalized Kubilius-Barban-Vinogradov bound for prime multiplicities

We present an assessment of the distance in total variation of \textit{arbitrary} collection of prime factor multiplicities of a random number in $[n]=\{1,\dots, n\}$ and a collection of independent geometric random variables. More precisely, we impose mild conditions on the probability law of the random sample and the aforementioned collection of prime multiplicities, for which a fast decaying bound on the distance towards a tuple of geometric variables holds. Our results generalize and complement those from Kubilius et al. which consider the particular case of uniform samples in $[n]$ and collection of "small primes". As applications, we show a generalized version of the celebrated Erdös Kac theorem for not necessarily uniform samples of numbers.

math.PR↗

arXiv2021

Limit Theorems for Additive Functionals of the Fractional Brownian Motion

We investigate first and second order fluctuations of additive functionals of a fractional Brownian motion (fBm) of the form \begin{align}\label{eq:abstractmain} Z_n=\left\{\int_{0}^{t}f(n^{H}(B_{s}-λ))ds\ ; t\geq 0 \right\} \end{align} where $B=\{B_{t}; t \geq 0\}$ is a fBm with Hurst parameter $H\in (0,1)$, $f$ is a suitable test function and $λ\in \mathbb{R}$. We develop our study by distinguishing two regimes which exhibit different behaviors. When $H\in(0,1/3)$, we show that a suitable renormalization of $Z_n$, compensated by a multiple of the local time of $B$, converges towards a constant multiple of the derivative of the local time of $B$. In contrast, in the case $H\in[1/3,1)$ we show that $Z_n$ converges towards an independent Brownian motion subordinated to the local time of $B$. Our results refine and complement those from the current literature and solve at the same time the critical case $H=1/3$, which had remained open until now.

math.PR↗