Some variations on a theme by P. de Jong
Date: 8 September 2022, Thursday
Time: 5pm AEST
Statistics
and Stochastic colloquium (part of the Colloquium Series of the
Department of Mathematics and Statistics) at La Trobe University jointly
organized with the Probability Victoria Seminar.
Contact the organizers: Kostya Borovkov kostya.borovkov@gmail.com, Andriy Olenko a.olenko@latrobe.edu.au
Speaker: Prof Giovanni Peccati (University of Luxembourg, Grand Duchy of Luxembourg)
Abstract:In a remarkable paper from 1990, the Dutch mathematician P. de Jong proved a striking Central Limit Theorem yielding that, for a sequence of normalized and degenerate U-statistics verifying a Lindeberg-type condition, convergence to Gaussian is equivalent to the convergence of their fourth cumulants to zero. Such a result is the ancestor of the collection of “fourth-moment theorems” for non-linear functionals of random fields, that have recently played a prominent role in several questions of mathematical physics and stochastic geometry. In my talk, I will first present some quantitative multidimensional extensions of de Jong’s result, obtained by using Stein’s method of exchangeable pairs. I will then discuss some recent functional versions of de Jong’s findings, both in the symmetric and non-symmetric cases. The results in the symmetric case yield some novel universality results for U-processes, generalizing a classic invariance principle by Miller and Sen (1972), and allowing one to establish a complete taxonomy of functional CLTs associated with counting statistics of random geometric graphs. My presentation is mainly based on the following references:
Ch. Döbler and G. Peccati: Quantitative de Jong Theorems in any dimension. EJP, 2016.
Ch. Döbler, M. Kasprzak and G. Peccati: Weak convergence of U-processes with size-dependent kernels. Ann. App. Prob., 2022
Ch. Döbler, M. Kasprzak and G. Peccati. The multivariate functional de Jong CLT. Probab. Th. Rel. Fields, 2022+
Zoom meeting link: https://unimelb.zoom.us/j/82317899187?pwd=TThhQmZrcGtxSGpQL2wzTHJjZlZjQT09
(if the above link doesn't work when you click it -- please copy & paste it into the address bar in your browser).
Password: 633070 (just in case)
A PDF file with the talk slides might become available for downloading from our seminar Webpage at https://probvic.wordpress.com/pvseminar/ prior to the talk (the above Zoom link will also be posted there shortly).
