Markov Chains with Asymptotically Zero Drift: Lamperti's Problem

Denis Denisov; Dmitry Korshunov; Vitali Wachtel

doi:10.1017/9781009554237

This text examines Markov chains whose drift tends to zero at infinity, a topic sometimes labelled as 'Lamperti's problem'. It can be considered a subcategory of random walks, which are helpful in studying stochastic models like branching processes and queueing systems. Drawing on Doob's h-transform and other tools, the authors present novel results and techniques, including a change-of-measure technique for near-critical Markov chains. The final chapter presents a range of applications where these special types of Markov chains occur naturally, featuring a new risk process with surplus-dependent premium rate. This will be a valuable resource for researchers and graduate students working in probability theory and stochastic processes.

- Aa Reduce text
- Aa Enlarge text

View selected items
Save to my bookmarks
Export citations
Download PDF (zip)
Save to Kindle
Save to Dropbox
Save to Google Drive
Save content to
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to .

To save content items to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Please be advised that item(s) you selected are not available.
You are about to save
Your Kindle email address

Please provide your Kindle email.

@free.kindle.com @kindle.com (service fees apply)

By using this service, you agree that you will only keep content for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services

Contents

Frontmatter
pp i-iv
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

Contents
pp v-viii
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

Preface
pp ix-xii
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

Notation and Conventions
pp xiii-xvi
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

1 - Introduction
pp 1-40
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

2 - Lyapunov Functions and Classification of Markov Chains
pp 41-87
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

3 - Down-Crossing Probabilities for Transient Markov Chain
pp 88-109
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

4 - Limit Theorems for Transient and Null Recurrent Markov Chains with Drift Proportional to 1/x
pp 110-153
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

5 - Limit Theorems for Transient Markov Chains with Drift Decreasing More Slowly Than 1/x
pp 154-190
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

6 - Asymptotics for Renewal Measure for Transient Markov Chain via Martingale Approach
pp 191-220
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

7 - Doob’s h-Transform: Transition from Recurrent to Transient Chain and Vice Versa
pp 221-235
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

8 - Tail Analysis for Recurrent Markov Chains with Drift Proportional to 1/x
pp 236-280
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

9 - Tail Analysis for Positive Recurrent Markov Chains with Drift Going to Zero More Slowly Than 1/x
pp 281-305
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

10 - Markov Chains with Asymptotically Non-Zero Drift in Cramér Case
pp 306-338
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

11 - Applications
pp 339-399
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

References
pp 400-407
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

Author Index
pp 408-409
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

Subject Index
pp 410-412
- - Get access
    
    Check if you have access via personal or institutional login
    
    Log in Register
- Export citation

Metrics

Total number of HTML views: 0

Total number of PDF views: 0 *

Loading metrics...

Total views: 0 *

Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.

Why this information is here

This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.

Accessibility Information

Accessibility compliance for the PDF of this book is currently unknown and may be updated in the future.

Markov Chains with Asymptotically Zero Drift

Lamperti's Problem

Book description

Refine List

Actions for selected content:

Contents

Frontmatter
pp i-iv

Contents
pp v-viii

Preface
pp ix-xii

Notation and Conventions
pp xiii-xvi

1 - Introduction
pp 1-40

2 - Lyapunov Functions and Classification of Markov Chains
pp 41-87

3 - Down-Crossing Probabilities for Transient Markov Chain
pp 88-109

4 - Limit Theorems for Transient and Null Recurrent Markov Chains with Drift Proportional to 1/x
pp 110-153

5 - Limit Theorems for Transient Markov Chains with Drift Decreasing More Slowly Than 1/x
pp 154-190

6 - Asymptotics for Renewal Measure for Transient Markov Chain via Martingale Approach
pp 191-220

7 - Doob’s h-Transform: Transition from Recurrent to Transient Chain and Vice Versa
pp 221-235

8 - Tail Analysis for Recurrent Markov Chains with Drift Proportional to 1/x
pp 236-280

9 - Tail Analysis for Positive Recurrent Markov Chains with Drift Going to Zero More Slowly Than 1/x
pp 281-305

10 - Markov Chains with Asymptotically Non-Zero Drift in Cramér Case
pp 306-338

11 - Applications
pp 339-399

References
pp 400-407

Author Index
pp 408-409

Subject Index
pp 410-412

Metrics

Full text views

Book summary page views

Accessibility standard: Unknown

Why this information is here

Accessibility Information

Markov Chains with Asymptotically Zero Drift

Lamperti's Problem

Book description

Refine List

Actions for selected content:

Save Search

Contents

Metrics

Full text views

Book summary page views

Accessibility standard: Unknown

Why this information is here

Accessibility Information