eCommons

 

Extremal Properties Of Markov Chains And The Conditional Extreme Value Model

Other Titles

Abstract

Multivariate extreme value theory has proven useful for modeling multivariate data in fields such as finance and environmental science, where one is interested in accounting for the tendency of observations to exceed an extremely high (or low) threshold. Recent work has developed extremal models by studying the conditional distribution of a random vector, conditional on one of the components becoming extreme. This provides a way to handle situations such as asymptotic dependence, where traditional techniques may be uninformative. In this thesis, we explore the implications of the assumption that such a conditional distribution is well approximated by a limiting probability distribution when the conditioning component is extreme. We consider a version of the conditional distribution specified by a transition function. If the transition kernel of a Markov chain satisfies our assumption, then a process known as the tail chain approximates the Markov chain over extreme states. We characterize the class of chains which admit such an approximation, and investigate the properties of the tail chain in relation to the distinction between extreme and non-extreme states. We find that, in general, the tail chain approximates a portion of the original process we term the "extremal component". We further derive the limit in distribution of a point process consisting of normalized Markov chain observations, expressing the limit in terms of the tail chain. We also consider the case where a transition function satisfying our assumption describes the dependence structure of a random vector. We establish conditions under which a conditional extreme value model is appropriate, and derive the form of the limiting measure.

Journal / Series

Volume & Issue

Description

Sponsorship

Date Issued

2012-08-20

Publisher

Keywords

Extreme Value Theory; Markov Chains; Point Processes

Location

Effective Date

Expiration Date

Sector

Employer

Union

Union Local

NAICS

Number of Workers

Committee Chair

Resnick, Sidney Ira

Committee Co-Chair

Committee Member

Nussbaum, Michael
Samorodnitsky, Gennady

Degree Discipline

Statistics

Degree Name

Ph. D., Statistics

Degree Level

Doctor of Philosophy

Related Version

Related DOI

Related To

Related Part

Based on Related Item

Has Other Format(s)

Part of Related Item

Related To

Related Publication(s)

Link(s) to Related Publication(s)

References

Link(s) to Reference(s)

Previously Published As

Government Document

ISBN

ISMN

ISSN

Other Identifiers

Rights

Rights URI

Types

dissertation or thesis

Accessibility Feature

Accessibility Hazard

Accessibility Summary

Link(s) to Catalog Record