A third-moment theorem and precise asymptotics for variations of stationary Gaussian sequences

In two new papers [2] and [8], sharp general quantitative bounds are given to complement the well-known fourth moment theorem of Nualart and Peccati, by which a sequence in a \u85xed Wiener chaos converges to a normal law if and only if its fourth cumulant converges to 0. The bounds show that the speed of convergence is precisely of order the maximum of the fourth cumulant and the absolute value of the third moment (cumulant). Specializing to the case of normalized centered quadratic variations for stationary Gaussian sequences, we show that a third moment theorem holds: convergence occurs if and only if the sequences third moments tend to 0. This is proved for sequences with general decreasing covariance, by using the result of [8], and \u85nding the exact speed of convergence to 0 of the quadratic variations third and fourth cumulants. [8] also allows us to derive quantitative estimates for the speeds of convergence in a class of log-modulated covariance structures, which puts in perspective the notion of critical Hurst parameter when studying the convergence of fractional Brownian motions quadratic variation. We also study the speed of convergence when the limit is not Gaussian but rather a second-Wiener-chaos law. Using a log-modulated class of spectral densities, we recover a classical result of Dobrushin-Major/Taqqu whereby the limit is a Rosenblatt law, and we provide new convergence speeds. The conclusion in this case is that the price to pay to obtain a Rosenblatt limit despite a slowly varying modulation is a very slow convergence speed, roughly of the same order as the modulation