Introduction to Markov processes, with coupling and convergence rates, and applications (MAGIC093)
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This course is part of the MAGIC core.
Markov processes in discrete and continuous time will be presented for elementary and general state spaces and for some jump & so called linear-Markov (semi-Markov) processes in continuous times. Standard stuff will include generators, Dynkin’s formula, ergodicity for finite state spaces, strong Markov, Feller & strong Feller properties. A bit more advanced material will include coupling & recurrence applied to convergence rates and to queueing & reliability systems.
Autumn 2017 (Monday, October 9 to Friday, December 15)
Some basic knowledge about Markov chains is highly desirable.
1. Stochastic processes. Definitions of a Markov process. 2. Examples: Random Walks. Generators. Chapman-Kolmogorov equations. 3. Dynkin’s identity. Stopping times, strong Markov property, Feller & strong Feller processes. 4. Irreducible Markov processes, ergodic theorem for finite state spaces and in general case. 5. Doob-Doeblin's and Markov-Dobrushin's conditions of ergodicity. 6. Positive & null recurrent, polynomially & exponentially recurrent Markov processes. 7. Coupling method, lemma about three random variables, application to convergence rates. 8. Applications to queueing: Erlang telephone systems, stationary regimes, convergence rates. 9. Piecewise-linear Markov processes, extended Erlang formulae & convergence rates. 10. Applications to some reliability theory problems.
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