The celebrated Breuer-Major Theorem (1983) states a CLT for rescaled sums of stationnary sequences of the form f(X_i), where the X_i are Gaussian, not necessarily independent, but with correlations decaying appropriately fast. Functional strenghtenings of this result were later obtained in several articles, one of the sharpest results being derived by Nourdin-Nualart (2020), where an invariance principle is derived under Lp integrability condition on f, for any p>2.
I shall present a rough path strengthening of these results showing that, under appropriate differentiability and integrability assumptions on a vector-valued function f, the rescaled sums of the variables f(X_i) together with their iterated sums converge in law to the distribution of a Brownian rough path. This is joint work with Tom Klose (University of Oxford) and Nicolas Perkowski (Freie Universität Berlin).
