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

Abstract

In two new papers (Biermé et al., 2013) and (Nourdin and Peccati, 2015), sharp general quantitative bounds are given to complement the wellknown fourth moment theorem of Nualart and Peccati, by which a sequence in a fixed 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 (Nourdin and Peccati, 2015), and finding the exact speed of convergence to 0 of the quadratic variation's third and fourth cumulants. (Nourdin and Peccati, 2015) 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.1991 Mathematics Subject Classification. 60G15, 60F05, 60H07, 60G22.

AR(1) processes driven by second-chaos white noise: Berry–Esséen bounds for quadratic variation and parameter estimation

“…The bound (26) is then a direct consequence of inequality (28) and the estimates given respectively in (29), (30), (34) and Theorem 6.…”

“…The main underlying theoretical result we draw on is the optimal estimation of totalvariation distances between chaos variables and the normal law, established in [31]. It was used for quadratic variations of stationary sequences in the Gaussian case in [29]. But when the Gaussian setting is abandonned, the result in [31] cannot be used directly.…”

“…where l 1 , l 2 , are defined in the previous section in (16), (19), and C 1 , C 2 , C 3 , C 4,r , r = 1, 2, 3 and C 5 are given in the lemmas below, respectively in (15), (18), (29), (30) and (34).…”

Optimal rates for parameter estimation of stationary Gaussian processes

“…A now classical result of Davydov and Martinova [9] was revived in recent years in [5,20] to estimate total-variation distances to G (H) ∞ . This can be achieved in our context as well, though for the sake of conciseness, we omit this study, only stating two basic results here, whose proofs would proceed as in [20] and [5] respectively. 1.…”

“…One may choose a high-degree polynomial f q such that the constant d 2 fq,2 (Z) may be much smaller than C 2 (Z); in this case, the dominant term in (20), which is for k = 1, has the same behavior in terms of n as for q = 2, but would be minimized because of the presence of the small multiplicative factor d 2 fq,2 (Z). If the other constants d 2 fq,2k (Z) for k 2 were much larger, this would have little effect for large n since they would be multiplicative of the asymptotically negligible tails |j|>n r Z (j) 2k in (20). In other words, for large q, the speed of convergence in (20) can be controlled by choosing f q with a small contribution to the term corresponding to H 2 in the Hermite polynomial decomposition (4).…”