Markov Chains with Asymptotically Zero Drift
Lamperti's Problem

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

Denis Denisov (Author), Dmitry Korshunov (Author), Vitali Wachtel (Author)

9781009554220, Cambridge University Press

Hardback, published 8 May 2025

428 pages
23.5 x 15.8 x 2.3 cm, 0.76 kg

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.

Subject Areas: Probability & statistics [PBT]