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 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.
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