Probability and Statistics seminar. "Non-homogeneous random walks in critical regimes". Mikhail Menshikov Durham University, UK
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In this talk we describe some recent joint work on non-classical random walks, focusing on
(i) zero-drift Markov chains in Euclidean spaces, which increment covariance function varies in such a way, that the walk can be recurrent (or transient) in any dimension d ≥ 2;
(ii) perturbations of zero-drift random walks, including the connection to the classical work of Lamperti;
(iii) an extension of the Lamperti problem to the case of heavy-tailed increments, including perturbations of martingales with infinite variance increments;
(iv) self-interacting processes: random walks that avoid their past convex hull.
The techniques are based on the Lyapunov functions, which allows us to address general questions of recurrence, transience, ballisticity, and so on.