Discrete and Continuous Models in probability
Period of duration of course
The second-year course “Elementi di Probabilità” is an obvious prerequisite. Moreover, familiarity will be assumed with several topics that are better dealt with in the third-year course named “Probabilità” of the bachelor’s degree in Mathematics at the University of Pisa, which runs in parallel to this one. Since the “Probabilità” course is lectured during the first semester, we are deferring the start of this course to November, so as to sync the two and rely on the “Probabilità” material with the correct timing.
- Review on Markov chains
- Definition and first properties of continuous time Markov processes with jumps
- Infinitesimal generator, examples
- The construction of jump Markov processes
- A few elements on martingales
- Dynkin formula, applications
- Entropy, convergence to equilibrium, spectral radius.
- Basic definitions and preliminary results, including the contraction principle
- Basic results for independent random variables, including Chernoff bounds
- Entropy and large deviations.
- Couplings and the total variation distance
- Markov chain convergence and mixing time
- Examples of using couplings to upper bound mixing time
- Path coupling (and the mixing time of SSEP)
- Some lower bounds, also using Chernoff bounds (Varopoulos-Carne)
Possible extras: spectral methods and the relaxation time; non-Markovian couplings; the coupling from the past technique; mixing in continuous time.
- Branching processes and their survival probability
- Total progeny and generation sizes
- The random walk interpretation
- Supercriticality and the Kesten-Stigum Theorem
- Poisson, binomial, geometric branching processes and applications to combinatorial trees
- Applications to random graphs: Erdős–Rényi, its phase transition and the giant component
Possible extras: size-biasing and local limits; criticality and scaling limits; more about Erdős–Rényi (threshold functions using large deviation estimates).
Notes given by the teachers