Discrete and Continuous Models in probability

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Written and oral exam


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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.


Markov processes
  1. Review on Markov chains
  2. Definition and first properties of continuous time Markov processes with jumps
  3. Infinitesimal generator, examples
  4. The construction of jump Markov processes
  5. A few elements on martingales
  6. Dynkin formula, applications
  7. Entropy, convergence to equilibrium, spectral radius.
Possible extrasan example of scaling limit. 
Large deviations
  1. Basic definitions and preliminary results, including the contraction principle
  2. Basic results for independent random variables, including Chernoff bounds
  3. Entropy and large deviations.
Possible extras: large deviations and statistical mechanics.
Couplings & mixing
  1. Couplings and the total variation distance
  2. Markov chain convergence and mixing time
  3. Examples of using couplings to upper bound mixing time
  4. Path coupling (and the mixing time of SSEP)
  5. Some lower bounds, also using Chernoff bounds (Varopoulos-Carne)
    Possible extrasspectral methods and the relaxation time; non-Markovian couplings; the coupling from the past technique; mixing in continuous time.
Branching processes
  1. Branching processes and their survival probability
  2. Total progeny and generation sizes
  3. The random walk interpretation
  4. Supercriticality and the Kesten-Stigum Theorem
  5. Poisson, binomial, geometric branching processes and applications to combinatorial trees
  6. Applications to random graphs: Erdős–Rényi, its phase transition and the giant component
    Possible extrassize-biasing and local limits; criticality and scaling limits; more about Erdős–Rényi (threshold functions using large deviation estimates).

Educational aims

The aim of the course is to present some fundamental themes in Probability that act as a complement to those traditionally taught at the University of Pisa (during both bachelor’s and master’s degrees).
A first topic we will cover is that of Markov processes with a finite or countable state space. Basic knowledge of discrete-time Markov chains will be assumed for third-year students, but the fundamentals will be briefly recalled at the beginning of the course. From here, we shall go into more advanced material about Markov chains themselves, as well as Markov processes with jumps in continuous time. 
A second central theme will be that of large deviations. While pursuing a Mathematics degree, students encounter limit theorems such as laws of large numbers and the central limit theorem. The theory of large deviations is a third important branch for limit theorems in Probability, and boasts applications to all parts of the discipline. We shall discuss the basics and present some more specific results, such as Chernoff bounds.
The next part of the course will present a series of further topics involving the theory developed thus far, with an eye for discrete models, random graphs and statistical mechanics. We shall introduce the mixing time of Markov chains, deal with various related problems, and present techniques to find upper and lower bounds for it. We will then develop part of the theory of branching processes and discuss some of their surprising applications to combinatorics and random graphs.

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