¡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.

Department of mathematics of the universities of Luxembourg and Singapore.

Postdoctoral 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.

2014-2018

Department of mathematics, University of Kansas, United States

Ph.D. student.
Research work in stochastic analisis and Stein-Malliavin method, under the supervision of David Nualart.

2008-20014

Department of Probability and Statistics, Research Center of Mathematics (CIMAT), Mexico.

M.Sc. student. Thesis oriented to Malliavin calculus and elaborated under the supervision of Juan Carlos Pardo.

Research

Malliavin calculus of Gaussian processes

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.

Centered 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.

Process 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.

Collection 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.

My 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.

Random matrices

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.

Quantitative limit theorems via relative log-concavity .

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.

Functional limit theorem for the self-intersection local time of the fractional Brownian motion.

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.

Limit theorems for linear statistics of matrix-valued gaussian processes

Seminario Mexico-Japon, 2022. A presentation of limit theorems for the linear statistics associated to the spectrum of matrix-valued Gaussian processes.

Eigenvalue collision for matrix Gaussian processes

Simposio de probabilidad y procesos estocásticos, UNAM 2017.
We provide conditions for the non-collision of the eigenvalues of matrix-valued Gaussian processes.

Organization of: XX escuela de probabilidad y estadistica

Departament 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.

Escuela 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.

Organization of: XIX escuela de probabilidad y estadistica

Departamento 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.

Conferencia temática "Un primer acercamiento a la investigacion en probabilidad"

Asociacion Mexiquense de Matematica Educativa, 2019. I discuss some of my personal experience regarding the task of conducting research activities in probability and statistics.