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MATH 4104: Advanced probability theory


1) Conditioning


Conditional expectation, conditional laws.


2) Discrete martingales.


Filtrations. Martingales, sub-martingales, super-martingales.

Stopping time, sigma algebra, stopping theorem.

Inequality of the number of Doob rises.

A.s. convergence theorem.

Doob’s maximal inequality.

Convergence in Lp (p>1), special case of L2 martingales.

Uniform integrability, convergence in L1. Retrograde martingales. Applications.


3) Markov chains.


Definitions of a Markov chain (countable state space and discrete-time), weak and strong Markov property.

Classification of states, irreducibility, transience, recurrence.

Invariant measure: existence, uniqueness.

Convergence theorems: ergodic theorem, convergence in law towards the invariant measure.

Reversible Markov chains, special case of Markov chains on a finite set.

Examples: Random walks, branching process, Ehrenfest model, Monte Carlo method, simple cut-off cases.


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