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 wellknown fourth moment theorem of Nualart and Peccati, by which a sequence in a …xed 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 sequence’s third moments tend to 0. This is proved for sequences with general decreasing covariance, by using the result of [8], and …nding the exact speed of convergence to 0 of the quadratic variation’s 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 motion’s 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. 2010 MSC: 60G15, 60F05, 60H07, 60G22 Keywords and phrases: stationary Gaussian process, Wiener space, central limit theorem, BerryEsséen, Breuer-Major, second chaos. 1 Introduction We are inspired by the following reformulation of Theorem 1.2 in [8], which is itself based on ideas contained in [2]. Theorem 1 (4 moment theorem in total variation and convergence rates). If (Fn)n 0 is a sequence in a …xed Wiener chaos (e.g. in the second chaos), and V ar [Fn] = 1, then (Fn)n 0 converges in law towards N (0; 1) if and only if E[F 4 n ]! 3 = E[N ], where N N (0; 1). Moreover the convergence rate in this case is Mn := max(E[F 4 n ] 3; E[F 3 n ] ), in the sense of commensurability for the total variation metric dTV (Fn; N) Mn, i.e. 9c; C > 0 : cMn dTV (Fn; N) CMn: (1) The …rst part of this theorem is known as the 4th moment theorem, proved originally by Nualart and Peccati in [13]. The second part, i.e. relation (1), suggests that the third moment is just as important as the 4th moment when investigating the normal convergence of sequences in a …xedWiener chaos. Theorem