Markov Chains with Asymptotically Zero Drift: Lamperti's Problem [Hardcover]

A comprehensive presentation of results for time-homogeneous Markov chains with asymptotically zero drift.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 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.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 analylS*