Fourth Moment Theorems for Markov Diffusion Generators

Inspired by the insightful article arXiv:1210.7587, we revisit the Nualart-Peccati-criterion arXiv:math/0503598 (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion generators. We are not only able to drastically simplify all of its previous proofs, but also to provide new settings of diffusive generators (Laguerre, Jacobi) where such a criterion holds. Convergence towards gamma and beta distributions under moment conditions is also discussed.Comment: 15 page

Optimal Convergence Rates and One-Term Edgeworth Expansions for Multidimensional Functionals of Gaussian Fields

We develop techniques for determining the exact asymptotic speed of convergence in the multidimensional normal approximation of smooth functions of Gaussian fields. As a by-product, our findings yield exact limits and often give rise to one-term generalized Edgeworth expansions increasing the speed of convergence. Our main mathematical tools are Malliavin calculus, Stein's method and the Fourth Moment Theorem. This work can be seen as an extension of the results of arXiv:0803.0458 to the multi-dimensional case, with the notable difference that in our framework covariances are allowed to fluctuate. We apply our findings to exploding functionals of Brownian sheets, vectors of Toeplitz quadratic functionals and the Breuer-Major Theorem

Four moments theorems on Markov chains

We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumptionwemake on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions. For elements of a Markov chaos, this bound can be reduced to just the first four moments.First author draf

Four moments theorems on Markov chaos

We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carr\'e du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions. For elements of a Markov chaos, this bound can be reduced to just the first four moments.Comment: 24 page

Limit theorems for general functionals of Brownian local times

In this paper, we present the asymptotic theory for integrated functions of increments of Brownian local times in space. Specifically, we determine their first-order limit, along with the asymptotic distribution of the fluctuations. Our key result establishes that a standardized version of our statistic converges stably in law towards a mixed normal distribution. Our contribution builds upon a series of prior works by S. Campese, X. Chen, Y. Hu, W.V. Li, M.B. Markus, D. Nualart and J. Rosen \cite{C17,CLMR10,HN09,HN10,MR08,R11,R11b}, which delved into special cases of the considered problem, such as quadratic, cubic and polynomial cases. We establish the limit theorem for general functions that satisfy mild smoothness and growth conditions. This extends the scope beyond the polynomial cases studied in previous works, providing a more comprehensive understanding of the asymptotic properties of the considered functionals

Four moments theorems on Markov chaos

We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumptionwe make on the Pearson distribution is that it admits four moments. These results are obtained by rst proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diusion generator and invariant measures of diusions, which is of independent interest, and making use of the new concept of chaos grade. For the heavy-tailed Pearson distributions, this seems to be the rst time that sucient conditions in terms of (nitely many) moments are given in order to converge to a distribution that is not characterized by its moments

Aplicación de requerimientos funcionales en la accesibilidad del hospital de la policía nacional del Perú de la ciudad de Trujillo

RESUMEN La presente tesis propone el diseño arquitectónico del Hospital de la Policía Nacional del Perú; cuyo objetivo es determinar los requerimientos funcionales con la accesibilidad; está estructurada de tal forma que permita conocer el impacto que el diseño, basado en la relación de las variables mencionadas, puede tener sobre el usuario en cuanto a su correcto desplazamiento en el recinto. Para ello, la investigación utiliza información relevante para el análisis de las variables; desarrolla un marco teórico en base a antecedentes encontrados para ser aplicado en el diseño arquitectónico del proyecto. Producto de esta investigación, se determinaron los criterios para el diseño como: zonificación, función, forma y escala en relación a la variable de requerimientos funcionales y los flujos de circulación necesaria, en relación a la variable de accesibilidad.; variables características de una arquitectura hospitalaria. Para esta propuesta se tuvo definido el terreno adecuado ubicado en el distrito de Trujillo. Finalmente, los resultados determinaron la relación directa entre las variables de estudio, como principios de la arquitectura hospitalaria, así como proponer la reducción de las 7 a 4 circulaciones para mejorar el flujo de recorrido en el Hospital de la Policía Nacional del Perú enmarcado en la provincia de Trujillo.ABSTRACT The present thesis, proposes the arquitect desing of the National Police Hospital of Peru, whose objetive is to stablish the funtional requirements with the accesibility, estructure in a way that allows to know the impact that the desing, based on the relation with the variables mentioned, can have on the user in terms of his correct displacement on the compound. For that, the investigation uses the relevant information for the analisis of the variables; develops a theorical framework based on the antecedents found to be aply on the arquitect desing of the proyect. Product of this investigation, the criteria for the desing was defined like: Zonification, funtion, shape and scale in relation with the funtional requirements variable and the needed circulation flow, in relation of the accesibility variable; caracteristic variables of a hospital arquitecture. For this proposal the suitable land was defined located on the Trujillo district. Finally, the results defined the direct relation betwen the study variables, as principles of hospital arquitecture, like proposing the reduction of the 7 to 4 circulations to improve the flow on the National Police Hospital located on the Trujillo province

Functional Gaussian approximations in Hilbert spaces: the non-diffusive case

We develop a functional Stein-Malliavin method in a non-diffusive Poissonian setting, thus obtaining a) quantitative central limit theorems for approximation of arbitrary nondegenerate Gaussian random elements taking values in a separable Hilbert space and b) fourth moment bounds for approximating sequences with finite chaos expansion. Our results rely on an infinite-dimensional version of Stein’s method of exchangeable pairs combined with the so-called Gamma calculus. Two applications are included: Brownian approximation of Poisson processes in Besov-Liouville spaces and a functional limit theorem for an edge-counting statistic of a random geometric graph.First author draf