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Contributions to the asymptotic study of Hermite driven processes
Full text linkThis thesis consists of two parts.
Part I is an introduction to Hermite processes, Hermite random fields, Fisher information and to the papers constituting the thesis. More precisely, in Section 1 we introduce Hermite processes in a nutshell, as well as some of its basic properties. It is the necessary background for the articles [a] and [c]. In Section 2 we consider briefly the multiparameter Hermite random fields and we study some less elementary facts which are used in the article [b]. In section 3, we recall some terminology about Fisher information related to the article [d]. Finally, our articles [a] to [d] are summarised in Section 4.
Part II consists of the articles themselves: [a] T.T. Diu Tran (2017): Non-central limit theorem for quadratic functionals of Hermite-driven long memory moving average processes. Stochastic and Dynamics, 18, no. 4. [b] T.T. Diu Tran (2016): Asymptotic behavior for quadratic variations of nonGaussian multiparameter Hermite random fields. Under revision for Probability and Mathematical Statistics. [c] I. Nourdin, T.T. Diu Tran (2017): Statistical inference for Vasicek-type model driven by Hermite processes. Submitted to Stochastic Process and their Applications. [d] T.T. Diu Tran (2017+): Fisher information and multivariate Fouth Moment Theorem. Main results have already been obtained. It should be submitted soon
Recent developments around the Malliavin-Stein approach (Fourth moment phenomena via exchangeable pairs)
Full text linkPart I is a survey, part II is a collection of papers
ON SOME ASYMPTOTIC RESULTS ON FUNCTIONALS OF WEAKLY STATIONARY RANDOM FIELDS
Full text linkFunctionals of random fields have always been a central topic in probability theory,
since its inception as a subject of study. The latter include, among others, partial
sums of random variables and geometric quantities associated to random functions
on manifolds. In this thesis, we investigate the asymptotic probabilistic behaviour
of integral functionals of weakly stationary random fields on expanding Euclidean
domains, with a special focus on additive (or nonlinear) functionals of stationary
Gaussian fields.
In Chapter 1 we first introduce the main mathematical objects and tools encoun-
tered in this work, concluding with an overview of the state of the art and our new
contributions related to the main research questions of this thesis. The two main
questions are the following: first, as the integration domain expands, does a central
limit theorem hold? Second, given two expanding integration domains, what is the
asymptotic covariance between their integral functionals?
Chapter 2 contains the paper "Spectral central limit theorem for additive func-
tionals of isotropic and stationary Gaussian fields", written in collaboration with Ivan
Nourdin. In this chapter, we prove that a large class of additive functionals of station-
ary, isotropic Gaussian fields satisfies a central limit theorem if an easily verifiable
condition on the spectral measure holds. This result brings to light a new class of
"strongly correlated" Gaussian fields whose additive functionals satisfy a central limit
theorem. This fact contradicts the intuition forged in the last four decades, starting
from the seminal works by Breuer, Dobrushin, Major, Rosenblatt and Taqqu.
Chapter 3 contains the paper "Fluctuations of polyspectra in spherical and Eu-
clidean random wave models", written in collaboration with Francesco Grotto and
Anna Paola Todino. Our main result provides the variance rate of any additive func-
tional of Euclidean (Berry’s random wave model) and spherical random waves, a
problem that was left as a conjecture ten years ago. To do this, we exploit a relation
between random waves and Pearson’s random walks.
Chapter 4 contains the paper "Asymptotic covariances for functionals of weakly
stationary random fields". Here we compute the asymptotic covariances of integral
functionals of weakly stationary random fields on expanding domains under assump-
tions that encompass the ones in the literature, deriving an explicit formula that
involves the directional derivative of the cross covariogram of two domains.
Chapter 5 contains the preprint "Limit theorems for p-domain functionals of
stationary Gaussian fields", written in collaboration with Nikolai Leonenko, Ivan
Nourdin and Francesca Pistolato. In this chapter we consider more general families
of additive functionals, which we call p-domain functionals, including as a special
case spatio-temporal functionals and 1-domain functionals considered in the previous
chapters. In this setting, we are able (under suitable assumptions) to reduce the
study of p-domain functionals to that of some 1-domain functionals, explaining some
recent findings in the literature in a new light
Limit theorems with Malliavin calculus and Stein's method
Full text linkWe use recent tools from stochastic analysis (such as Stein's method and Malliavin calculus) to study the asymptotic behaviour of some functionals of a Gaussien Field
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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