2016/03/01 by Léo Neufcourt, Neufcourt, Leo, Frédéri Viens +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Financial Risk and Volatility Modeling #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1603.00365
In two new papers (Bierme et al., 2013) and (Nourdin and Peccati, 2015),\nsharp general quantitative bounds are given to complement the well-known\nfourth moment theorem of Nualart and Peccati, by which a sequence in a fixed\nWiener chaos converges to a normal law if and only if its fourth cumulant\nconverges to 0. The bounds show that the speed of convergence is precisely of\norder the maximum of the fourth cumulant and the absolute value of the third\nmoment (cumulant). Specializing to the case of normalized centered quadratic\nvariations for stationary Gaussian sequences, we show that a third moment\ntheorem holds: convergence occurs if and only if the sequence's third moments\ntend to 0. This is proved for sequences with general decreasing covariance,\nby using the result of (Nourdin and Peccati, 2015), and finding the exact speed\nof convergence to 0 of the quadratic variation's third and fourth cumulants.\n(Nourdin and Peccati, 2015) also allows us to derive quantitative estimates for\nthe speeds of convergence in a class of log-modulated covariance structures,\nwhich puts in perspective the notion of critical Hurst parameter when studying\nthe convergence of fractional Brownian motion's quadratic variation. We also\nstudy the speed of convergence when the limit is not Gaussian but rather a\nsecond-Wiener-chaos law. Using a log-modulated class of spectral densities, we\nrecover a classical result of Dobrushin-Major/Taqqu whereby the limit is a\nRosenblatt law, and we provide new convergence speeds. The conclusion in this\ncase is that the price to pay to obtain a Rosenblatt limit despite a slowly\nvarying modulation is a very slow convergence speed, roughly of the same order\nas the modulation.\n