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Student’s t-Distribution and Related Stochastic Processes

  • Book
  • © 2013

Overview

Authors:
  1. Bronius Grigelionis

    1. University of Vilnius, Vilnius, Lithuania

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  • In-depth study of the infinite divisibility and self-decomposability properties of central and noncentral Student’s distributions
  • Extreme value theory for such diffusions is developed
  • Flexible and statistically tractable Kolmogorov-Pearson diffusions are described
  • Includes supplementary material: sn.pub/extras

Part of the book series: SpringerBriefs in Statistics (BRIEFSSTATIST)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xi
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  2. Introduction

    • Bronius Grigelionis
    Pages 1-7

  3. Asymptotics

    • Bronius Grigelionis
    Pages 9-20

  4. Preliminaries of Lévy Processes

    • Bronius Grigelionis
    Pages 21-40

  5. Student-Lévy Processes

    • Bronius Grigelionis
    Pages 41-50

  6. Student OU-Type Processes

    • Bronius Grigelionis
    Pages 51-56

  7. Student Diffusion Processes

    • Bronius Grigelionis
    Pages 57-76

  8. Miscellanea

    • Bronius Grigelionis
    Pages 77-91

  9. Back Matter

    Pages 93-99
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Keywords

About this book

This brief monograph is an in-depth study of the infinite divisibility and self-decomposability properties of central and noncentral Student’s distributions, represented as variance and mean-variance mixtures of multivariate Gaussian distributions with the reciprocal gamma mixing distribution. These results allow us to define and analyse Student-Lévy processes as Thorin subordinated Gaussian Lévy processes. A broad class of one-dimensional, strictly stationary diffusions with the Student’s t-marginal distribution are defined as the unique weak solution for the stochastic differential equation. Using the independently scattered random measures generated by the bi-variate centred Student-Lévy process, and stochastic integration theory, a univariate, strictly stationary process with the centred Student’s t- marginals and the arbitrary correlation structure are defined. As a promising direction for future work in constructing and analysing new multivariate Student-Lévy type processes, the notion of Lévy copulas and the related analogue of Sklar’s theorem are explained.

Reviews

“It is self-contained and summarizes the most recent results by the author related to t-distribution and their processes. … Grigelionis has pulled together an excellent overview in Student t-distribution and processes, which has not previously been available. The book is written at a highly scholarly level and should appeal to those with an interest in applied probability methodology and applications. It should be for students who have had an advanced course in probability.” (Stergios B. Fotopoulos, Technometrics, Vol. 58 (3), August, 2016)

Authors and Affiliations

  • University of Vilnius, Vilnius, Lithuania

    Bronius Grigelionis

About the author

Prof. Grigelionis is a senior research fellow at the Institute of Mathematics and Informatics of Vilnius University, member of the Lithuanian Academy of Sciences and the International Statistical Institute. He has done extensive research in stochastic analysis and its applications. These include the semimartingale characterisation of stochastic processes with conditionally independent increments and solutions of stochastic Ito's equations, stochastic nonlinear filtering equations, optimal stopping of stochastic processes - joint research with A. Shiryaev - criteria of weak convergence of stochastic processes - joint research with R. Mikulevičius - etc. His current research topics are the properties of mixed Gaussian distributions and related stochastic processes.

Bibliographic Information

  • Book Title: Student’s t-Distribution and Related Stochastic Processes

  • Authors: Bronius Grigelionis

  • Series Title: SpringerBriefs in Statistics

  • DOI: https://doi.org/10.1007/978-3-642-31146-8

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Author(s) 2013

  • Softcover ISBN: 978-3-642-31145-1Published: 18 September 2012

  • eBook ISBN: 978-3-642-31146-8Published: 18 September 2012

  • Series ISSN: 2191-544X

  • Series E-ISSN: 2191-5458

  • Edition Number: 1

  • Number of Pages: XI, 99

  • Topics: Statistics, general

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