Academic

Let $X$ be a centered, unit-variance and stationary Gaussian sequence with covariance function $r$. Let $f$ be a square integrable function with respect to the standard Gaussian measure on the real line. Consider the sequence $F_n$ of partial sums associated with $f(X_n)$. The celebrated Breuer-Major theorem provides sufficient conditions on the covariance $r$ and of the Hermite index of $f$ in order for $F_n$ to exhibit Gaussian fluctuations. Surprisingly, and despite the fact this theorem has far-reaching applications in many different areas (including mathematical statistics, signal processing or geometry of random nodal sets), a necessary and sufficient condition implying the weak convergence in the space of càdlàg functions endowed with the Skorohod topology is still missing. We will explain in this talk how to fill this gap. This is a joint work with David Nualart (Kansas University).