¡Hola! My name is Arturo Jaramillo Gil. Since December of 2020 I work as researcher at the Center of Research in Mathematics (CIMAT), Guanajuato. In 2018 I obtained my PhD in mathematics at the university of Kansas. During this time, I had the great pleasure of conduction research in limit theorems and Maliavvin calculus in collaboration with David Nualart. From September 2018 to November 2020, I participated in a postdoctorate program jointly with the universities of Luxembourg and Singapore, in collaboration with the research groups of Ivan Nourdin, Giovanni Peccati, Adrian Roellin y Louis H.Y. Chen in topics of Malliavin calculus and Stein's method.
download cvPostdoctoral research in mathematics under the bilateral agreement between Luxembourg and Singapore, boosted by Fonds National de la Recherche. Collaborated with the groups of Ivan Nourdin, Giovanni Peccati, Adrian Roellin y Louis H.Y. Chen in topics of Malliavin calculus, limit theorems and Stein's method.
Ph.D. student. Research work in stochastic analisis and Stein-Malliavin method, under the supervision of David Nualart.
M.Sc. student. Thesis oriented to Malliavin calculus and elaborated under the supervision of Juan Carlos Pardo.
Study of differential operators for Gaussian processes.
Malliavin calculus (also known as variational claculus in the Wiener space) is an infinite dimenional differential calculus in the Wiener space. Its applications include the sutdy estudio of anticipating stochastic integrals, the regularity of functionals of Gaussian processes, properties of the fractional Brownian motion and limit theorems for functionals of Gaussian processes.
Check my projects:
Asymptotic properties of the derivative of self-intersection local time of fractional Brownian motion Symmetric stochastic integrals with respect to a class of self-similar Gaussian processes Functional limit theorem for the self-intersection local time of the fractional Brownian motion Convergence of the empirical spectral distribution of Gaussian matrix-valued processesCentered Gaussian process with self similarity parameter H with stationary increment that generalizes the classical Brownian motion.
The fractional Brownian motion of Hurst parameter H is a self-similar Gaussian process with stationary increments and self-similarity parameter H. This process is very attractive from the modelling point of view, as an adequate tuning of the Hurst parameter allows us to typically obtain a good approximation of real-life phenomena. My main interests areas in this topic include the study of high frequency statistics, tutuations of stochastic integrals and the study of the asymptotic spectrum of matrix-valued processes.
Check my projects:
Approximation of Fractional Local Times: Zero Energy and Derivatives Asymptotic properties of the derivative of self-intersection local time of fractional Brownian motion Symmetric stochastic integrals with respect to a class of self-similar Gaussian processes Functional limit theorem for the self-intersection local time of the fractional Brownian motion Convergence of the empirical spectral distribution of Gaussian matrix-valued processesProcess that measures the amount of time that a stochastic process spends at a given level.
The local time at level y of a fractional Brownian motion X is a random variable that measures the amount of time that the process X spends around y. I am interested in the study of local times of X as well and their associated derivatives, as well as the applications to the study of high-frequency statistics. Additionally, I am interested in the study of the self-intersection local time of the fractional Brownian motion, which measures the amount of time that the trajectories of X intersect themselves.
Check my projects:
Approximation of Fractional Local Times: Zero Energy and Derivatives Asymptotic properties of the derivative of self-intersection local time of fractional Brownian motion Symmetric stochastic integrals with respect to a class of self-similar Gaussian processes Functional limit theorem for the self-intersection local time of the fractional Brownian motionCollection of probabilistic techniques that allow us to estimate the distance between probability measures by means of differential operators.
Stein's method denotes a collection of probabilistic techniques that allows us to estimate the distance between probability measures by means of differential operators. I am interested in the application of Stein's method to random matrices, probabilistic number theory and limit theorems in the Wiener space. Additionally, I have conducted research on the so called fourth moment phenomena, a very interesting technique for esimating probability distances of standarized variables by means of moments of order fourth. Another of my areas of interest consists on developing and extending the theory Stein' s method of non-Gaussian random variables, such as the semicircular law and the Wishart distribution.
Check my projects:
Convergence of the Fourth Moment and Infinite Divisibility: Quantitative estimatesMy main research areas are analysis in the Wiener space, limit theorems, fractional Brownian motion, Stein's method, local times, random matrices and probabilistic number theory.
Study of the spectrum of random matrices with adequate symmetries.
Another of my research areas consists of the study of the asymptotic properties of the spectrum of random matrices by means of Malliavin calculus, with particular emphasis in the case where the entries of the underlying matrices are stochastic processesin the Wiener space.
Check my projects:
Convergence of the empirical spectral distribution of Gaussian matrix-valued processesArturo Jaramillo, Chiara Amorino, Mark Podolskij. Preprint, 2023
We establish mixed Gaussian limit theorems for the fluctuations for partially observed high-frequency statistics for Levy processes.
Arturo Jaramillo, James Melbourne. Preprint, 2022
We study limit theorems for log-concave probability measures. As applications, we get estimations of classical limit theorems such as the law of rare events, binomial Poisson approximations, as well as some more modern ones such as limit theorems for random matroids and intrinsic volumes.
Chiara Amorino, Arturo Jaramillo, Mark Podolskij. Preprint, 2022
We study non-central limit theorems for the optimal estimator of the local time and occupation local time for symmetric alpha stable processes.
Louis H. Y. Chen, Arturo Jaramillo, Xiaochuan Yang. Preprint, 2021
We determine the quantitative asymptotic behavior of the p-valuations of samples of numbers in 1,...,n, under general condition.
A. Jaramillo, I. Nourdin, D. Nualart, G. Peccati. Preprint, 2021
We study the asymptotic behavior of additive functionals of the fractional Brownian motions for an arbitrary choice of the underlying Hurst parameter.
L.H.Y. Chen, A. Jaramillo, X. Yang. Preprint, 2021
We determine a generalized and quantitatiev version of the Erdos Kac theorem by means of Stein's method.
M. Diaz, A. Jaramillo, JC. Pardo. Annales de l'Institut Henri Poincare, Probabilites et Statistiques, 2021
We study the functional fluctuations of the spectrum of matrix-valued Gaussian processes.
A. Jaramillo, I. Nourdin, G. Peccati. Annals of Applied Probability, 2021
The derivatives of local times are introduced as a tool for studyinghigh-frequency statistics.
A. Jaramillo, D. Nualart. Random matrices: Theory and Applications (en proceso de publicación
A new methodology for determining the non-collision property for the eigenvalues of matrix-valued Gaussian processes is established.
A. Jaramillo, JC. Pardo, JL Pérez. Electronic Journal of Probability (2019) 10. 22-
The asymptotic behavior of matrix-valued Gaussian processes is determined (regardless of the existence of collision of the associated eigenvalues).
A. Jaramillo, D. Nualart. Annales de l'institut Henri Poincaré (2019) 22,481-528
We establish a functional limit theorem for the self-intersection local time of the fractional Brownian motion. Additionally, we propose a new methodology for proving tightness of stochastic processes.
D. Harnett, A. Jaramillo, D. Nualart. Journal of Theoretical Probability (2019) 3, 1105-1144.
We establish asymptotic Gaussianity for the symmetric integrals associated to general self-similar Gaussian processes.
A. Jaramillo, D. Nualart. Stochastic Processes and Their Applications (2017) 127. 669-700.
We study the asymptotic behavior of the chaotic components of the derivative of the self-intersection local time for the fractional Brownian motion.
O. Arizmendi, A. Jaramillo. Electronic Communications in Probability (2014) 19, 1-12.
We give estimates on the Kolmogorov distance towards the standard Gaussian distribution for infinitely divisible random variables.
Stein Symposium, the golden anniversary, Singapur, 2022. Study of additive functions for uniform samples via Stein's method.
SlidesSeminario Mexico-Japon, 2022. A presentation of limit theorems for the linear statistics associated to the spectrum of matrix-valued Gaussian processes.
SlidesUniversidad de Costa Rica, 2021. We study the fluctuations of linear statistics associated to the spectrum of matrix-valued Gaussian processes.
SlidesSeminario hispanoparlante, 2021. We study Gaussian mixed limits for additive the functionals of the fractional Brownian motion.
SlidesSeminario de charlas cortas, CIMAT, 2021. We present some of the applications of Stein's method in diverse areas of mathematics.
SlidesUniversita degli Studi di Milano-Bicocca, 2020. We study arithmetic additive functions by means of stochastic analysis on the Poisson space.
SlidesITAM, 2020. We study arithmetic additive functions by means of stochastic analysis on the Poisson space.
SlidesBerlin Technische, 2020. We give a purely probabilistic proof of the celebrated Erdos Kac theorem.
SlidesBernoulli IMS Symposium 2020. We introduce the derivative of the local time as a tool for studying high frequency statistics.
SlidesNational University of Singapore, 2019. We study the fluctuations of the spectrum of matrix-valued Gaussian processes.
SlidesSimposio de probabilidad y procesos estocásticos, UNAM 2017. We provide conditions for the non-collision of the eigenvalues of matrix-valued Gaussian processes.
SlidesProbability Seminar, University of Kansas 2017. We study the functional behavior of the asymptotic spectrum of matrix-valued Gaussian processes.
DiapositivasCIMAT, 2022. We study the representation of Dirichlet problems by means of stochastic calculus techniques.
ConstanciaDepartament de probability and statistics, CIMAT, 2022. This event is directed to advanced students in bachelor and graduate students in mathematics, statistics and afine disciplines. It has as general objetive the dissemination of research of topics on active areas of research in probability and statistics.
LigaInstituto Tecnológico Superior de Salvatierra Salvatierra; CIMAT, 2022. We study the frequency of the first digit for certain samples of numbers.
ConstanciaEscuela de verano, CIMAT, 2021. We study the coupon collector problem and exhibit a vast family of practical problems that can be addressed with this theory.
ConstanciaDepartamento de probabilidad y estadistica, CIMAT, 2021. This event is directed to advanced students in bachelor and graduate students in mathematics, statistics and afine disciplines. It has as general objetive the dissemination of research of topics on active areas of research in probability and statistics.
LigaAsociacion Mexiquense de Matematica Educativa, 2019. I discuss some of my personal experience regarding the task of conducting research activities in probability and statistics.
Constancia SlidesWebpage devoted to the study of the celebrated Malliavin-Stein method (mantained by Ivan Nourdin). Link
Seminario semanal virtual de probabilidad, dirigido a la comunidad hispanohablante. Link
Virtual seminar on topics of Malliavin calculus on the Poisson space. Link
Seminar devoted to unite researchers in probability on the globe. Link
Annual event for talking about mathematics in a friendly event. Link
Virtual seminar on elements of rough paths and its applications. Link
Seminar of probability, CIMAT. Link