Guangqu Zheng of Boston University : Generalized Donsker invariance principe in dependent setting

Posted by Cheng Ouyang , part of the Analysis and Applied Mathematics Seminar.

At

Oct. 5, 2026, 4 p.m.

In

636 SEO

Abstract

In this talk, we will recall the classical central limit theorem (CLT) and talk about a generalization of CLT for a stationary sequence X_k = f(G_k), where G is a centered stationary Gaussian sequence and f is square-integrable with respect to the standard Gaussian measure on reals. Motivated by the non-CLT results by Dobrushin, Major, Rosenblatt, and Taqqu, Peter Breuer and Peter Major, in their 1983 celebrated paper, established the asymptotic normality of the partial sum when the Gaussian covariance function has sufficient decay. Their work is on the convergence of finite-dimensional distributions, while the tightness problem (that would lead to a functional CLT) had remained open in its generality. Recent work by Nourdin and Nualart established tightness under the extra (2+) moment assumption by using Malliavin-Meyer argument. In a recent work with Mansanarez and Poly, we introduced a Gaussian prediction-theoretic framework to allow us to remove the extra moment assumption. The results in MPZ paper are applicable for `most’ Gaussian sequence and for all test function f in the finite variance setting. In a work with Dionysis Milesis, we finally remove all unnecessary assumptions.