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© 2013

Student’s t-Distribution and Related Stochastic Processes

Book

Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Bronius Grigelionis
    Pages 1-7
  3. Bronius Grigelionis
    Pages 9-20
  4. Bronius Grigelionis
    Pages 21-40
  5. Bronius Grigelionis
    Pages 41-50
  6. Bronius Grigelionis
    Pages 51-56
  7. Bronius Grigelionis
    Pages 57-76
  8. Bronius Grigelionis
    Pages 77-91
  9. Back Matter
    Pages 93-99

About this book

Introduction

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.

Keywords

Bessel function Gaussian Lévy process H-diffusion Self-decomposability Thorin subordinator

Authors and affiliations

  1. 1.University of VilniusVilniusLithuania

About the authors

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
  • Copyright Information The Author(s) 2013
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-642-31145-1
  • eBook ISBN 978-3-642-31146-8
  • Series ISSN 2191-544X
  • Series E-ISSN 2191-5458
  • Edition Number 1
  • Number of Pages XI, 99
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Statistics, general
  • Buy this book on publisher's site

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)