Some variations on a theme by P. de Jong

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

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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).