Stochastic Analysis Seminar

The Stochastic Analysis Seminar consists of three independent parts.

1. The first part is a series of seminars by outside speakers. See the speaker schedule for further information.
2. The second part is devoted to an informal seminar course on a contemporary topic in probability or stochastic analysis that is not currently taught in the curriculum. The current topic is detailed below.
3. The third part is a student-run seminar series (ORF 557/558).

The Stochastic Analysis Seminar is organized by Rene Carmona and Ramon van Handel.

Spring 2013: Random graphs

New feature this semester: stochastic analysis seminar blog.
Notes from the lectures and any announcements will be posted there.

Complex graphs and networks are ubiquitous throughout science and engineering, and possess beautiful mathematical properties. The goal of these informal lectures is to provide an introduction to some of the most basic results in the theory of (Erdos-Renyi) random graphs from the probabilistic viewpoint: chromatic number and clique number, and the size of the largest component in the subcritical, supercritical, and critical regimes. If time permits, we will discuss Brownian limits.

Prerequisites: Probability at the level of ORF 526 is assumed.

Time and location: Thursdays, 4:50-6:20, Bendheim Center classroom 103.
The first lecture will be on February 28. There will be no lecture on March 28 (colloquium).

References:

• N. Alon and J. Spencer, "The Probabilistic Method"
• R. Durrett, "Random Graph Dynamics"
• A. Nachmias and Y. Peres, "The critical random graph, with martingales", Israel J. Math. 176 (2010), p. 29-41
• D. Aldous, "Brownian excursions, critical random graphs and the multiplicative coalescent", Ann. Probab. 25 (1997), p. 812-854
• G. Grimmett, "Probability on Graphs"
• R. van der Hofstad, "Random Graphs and Complex Networks" (lecture notes)
• B. Bollobas, "Random Graphs"
• S. Janson, T. Luczak, A. Rucinski, "Random Graphs"

Fall 2012: Random matrices

Both the theory and applications of random matrices have attracted significant attention in recent years. In particular, much effort has gone into understanding the spectral properties of large random matrices, with applications ranging from statistical physics to signal processing to high-dimensional statistical inference.

The goal of these informal lectures is to provide a basic introduction to some selected results in random matrix theory. Potential topics include: Wigner matrices and the semicircle distribution; Wishart matrices and the Marcenko-Pastur distribution; asymptotics of the top and bottom singular values; fluctuations of linear eigenvalue statistics and of the top eigenvalue; Dyson Brownian motions and related topics. If time permits (which is unlikely) we will introduce some free probability.

Prerequisites: Probability at the level of ORF 526 is assumed.

Time and location: Thursdays, 4:30-6:00, Bendheim Center classroom 103.
The first lecture is on September 20. There will be no lecture on October 25 and November 1.

References:

• G. W. Anderson, A. Guionnet, O. Zeitouni, "An Introduction to Random Matrices"
• Z. Bai and J. W. Silverstein, "Spectral Analysis of Large Dimensional Random Matrices"
• T. Tao, "Topics in Random Matrix Theory"
• R. Vershynin, "Introduction to the non-asymptotic analysis of random matrices" (lecture notes)

Fall 2011: Concentration of measure

The law of large numbers states that the average of many independent random variables is close to its expectation. It turns out that this simple fact is a special case of a much more general phenomenon that could be informally phrased as follows: "a function of many independent random variables, that does not depend too much on any one of them, is nearly constant". This idea, called concentration of measure, appears in many different areas of probability and its applications, and there exists a powerful set of tools to establish that such properties hold in a precise quantitative sense (in the form of sharp nonasymptotic estimates on the deviation of the random variable of interest from its expectation or median).

The goal of these informal lectures is to provide a basic introduction to such methods. Potential topics include Chernoff bounds, Hoeffding, Bernstein and Azuma inequalities, bounded differences, Gaussian concentration, isoperimetry, log-Sobolev inequalities, transportation cost inequalities, Talagrand's concentration inequalities, and some random applications.

Prerequisites: Probability at the level of ORF 526 is assumed.

Time and location: Thursdays 4:30-5:30 PM in Bendheim Center classroom 103.
The first lecture is on September 15. There will be no lecture on October 6 and 27, and December 15.

References: The following references (ordered alphabetically) discuss theory and applications of concentration of measure from different perspectives. We may use parts of some of these references, but will not follow one particular reference throughout.

• N. Alon and J. Spencer, "The Probabilistic Method"
• N. Berestycki and R. Nickl, "Concentration of Measure" (lecture notes)
• D. Dubhashi and A. Panconesi, "Concentration of Measure for the Analysis of Randomised Algorithms"
• M. Ledoux, "The Concentration of Measure Phenomenon"
• G. Lugosi, "Concentration-of-measure inequalities" (lecture notes)
• P. Massart, "Concentration Inequalities and Model Selection" (St Flour lecture notes)
• C. McDiarmid, "Concentration"
  C. McDiarmid, "On the method of bounded differences"
• G. Pisier, "Probabilistic methods in the geometry of Banach spaces"
• J. M. Steele, "Probability Theory and Combinatorial Optimization"
• M. Talagrand, "Concentration of measure and isoperimetric inequalities in product spaces"
  M. Talagrand, "New concentration inequalities in product spaces"
  M. Talagrand, "A new look at independence"