This text provides a comprehensive presentation of results for time-homogeneous Markov chains with asymptotically zero drift. Including novel results and original research, this monograph will interest researchers and graduate students in probability, statistics and their applications.
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.
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A comprehensive presentation of results for time-homogeneous Markov chains with asymptotically zero drift.
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 slower 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 slower than 1/x;
10. Markov
chains with asymptotically non-zero drift in Cramér's case;
11. Applications.
Denis Denisov is Reader in Probability in the Department of Mathematics at the University of Manchester. His research interests include multidimensional random walks, ordered and conditioned random walks, Markov chains and diffusion processes. Dmitry Korshunov is Professor in the School of Mathematical Sciences at Lancaster University. He is an expert in stochastic processes, Markov chains, large deviation theorems, heavy tail phenomena, and limit theorems. He previously co-authored An Introduction to Heavy-Tailed and Subexponential Distributions (2011, 2013). Vitali Wachtel is Professor for Mathematics at the University of Bielefeld. His research interests include, besides Markov chains, exit times and conditional distributions for multidimensional random walks, branching processes and large deviations.
