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
1 - Introduction
Published online by Cambridge University Press: aN Invalid Date NaN
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
Summary
In Introduction we mostly discuss nearest neighbour Markov chains which represent one of the two classes of Markov chains whose either invariant measure in the case of positive recurrence or Green function in the case of transience is available in closed form. Closed form makes possible direct analysis of such Markov chains: classification, tail asymptotics of the invariant probabilities or Green function. This discussion sheds some light on what we may expect for general Markov chains. Another class is provided by diffusion processes which are also discussed in Introduction.
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- Markov Chains with Asymptotically Zero DriftLamperti's Problem, pp. 1 - 40Publisher: Cambridge University PressPrint publication year: 2025