I will review the construction of a Wishart matrix and the moments of it eigenvalue distribution. In the following weeks I will show how to find the distribution of the large N limit and interpret it in terms of non-crossing partitions. I will keep the exposition elementary and accessible.
Notes
Tuesday, October 4, 11:00 - 12:00, Jeff 202
Moments of complex Gaussian random vectors
Contining the discussion from last week, we will discuss Gaussian random vectors and their moments.
Tuesday, September 27, 11:00 - 12:00, Jeff 202
Moments and cumulants of random variables, Part II
In our previous meeting we talked about the moments and the cumulants associated to a random variable. We introduced the cumulants from two approaches, in one hand an analytic approach, while in the other hand a more combinatorial one. For this talk, we will prove these two definitions are equivalents. We will talk in further detail about the combinatorial definition and see how the moment-cumulant relation can be achieved via the mobius inversion theorem. We will see how it is possible to extend the cumulants to multilinear functions and recast a multilinear version of the moment-cumulant relation. If time permits, we will introduce some matrix models of interest in Random Matrix Theory.
Tuesday, September 20, 11:00 - 12:00, Jeff 202
Moments and cumulants of random variables, Part I
Given a random variable with all its moments, it is possible to expand the logarithm of its characteristic function as a power series. The coefficients on this series are called the cumulants of the random variable. Cumulants turns out to be a quite interesting object as they describe properties that the moments may not. I will introduce the concept of cumulants, or “classical cumulants”, from two different perspectives, in one hand a complete analytic definition while in the other hand a more combinatorial definition, as we will check, those two definitions are equivalent. We will compute these cumulants for popular cases such as the Gaussian case. Most of the talk will be based in section 1.1 “Moments and cumulants of random variables” from “Free probability and random matrices” by Mingo and Speicher, while some other results may be based on “Lectures on the combinatorics of free probability” by Nica and Speicher.