Book contents
- Frontmatter
- Contents
- Preface
- Notation and Conventions
- 1 Introduction
- 2 Lyapunov Functions and Classification of Markov Chains
- 3 Down-Crossing Probabilities for Transient Markov Chain
- 4 Limit Theorems for Transient and Null Recurrent Markov Chains with Drift Proportional to 1/x
- 5 Limit Theorems for Transient Markov Chains with Drift Decreasing More Slowly Than 1/x
- 6 Asymptotics for Renewal Measure for Transient Markov Chain via Martingale Approach
- 7 Doob’s h-Transform: Transition from Recurrent to Transient Chain and Vice Versa
- 8 Tail Analysis for Recurrent Markov Chains with Drift Proportional to 1/x
- 9 Tail Analysis for Positive Recurrent Markov Chains with Drift Going to Zero More Slowly Than 1/x
- 10 Markov Chains with Asymptotically Non-Zero Drift in Cramér Case
- 11 Applications
- References
- Author Index
- Subject Index
9 - Tail Analysis for Positive Recurrent Markov Chains with Drift Going to Zero More Slowly Than 1/x
Published online by Cambridge University Press: aN Invalid Date NaN
- Frontmatter
- Contents
- Preface
- Notation and Conventions
- 1 Introduction
- 2 Lyapunov Functions and Classification of Markov Chains
- 3 Down-Crossing Probabilities for Transient Markov Chain
- 4 Limit Theorems for Transient and Null Recurrent Markov Chains with Drift Proportional to 1/x
- 5 Limit Theorems for Transient Markov Chains with Drift Decreasing More Slowly Than 1/x
- 6 Asymptotics for Renewal Measure for Transient Markov Chain via Martingale Approach
- 7 Doob’s h-Transform: Transition from Recurrent to Transient Chain and Vice Versa
- 8 Tail Analysis for Recurrent Markov Chains with Drift Proportional to 1/x
- 9 Tail Analysis for Positive Recurrent Markov Chains with Drift Going to Zero More Slowly Than 1/x
- 10 Markov Chains with Asymptotically Non-Zero Drift in Cramér Case
- 11 Applications
- References
- Author Index
- Subject Index
Summary
In Chapter 9 we consider a recurrent Markov chain possessing an invariant measure which is either probabilistic in the case of positive recurrence or σ-finite in the case of null recurrence. Our main aim here is to describe the asymptotic behaviour of the invariant distribution tail for a class of Markov chains with asymptotically zero drift going to zero more slowly than 1/x. We start with a result which states that a typical stationary Markov chain with asymptotically zero drift always generates a heavy-tailed invariant distribution which is very different from the case of Markov chains with asymptotically negative drift bounded away from zero. Then we develop techniques needed for deriving precise tail asymptotics of Weibullian type.
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- Information
- Markov Chains with Asymptotically Zero DriftLamperti's Problem, pp. 281 - 305Publisher: Cambridge University PressPrint publication year: 2025