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Abstract: Hermite processes are self-similar processes with stationary increments which appear as limits of normalized sums of random variables with long range dependence. The Hermite process of order 1 is fractional Brownian motion and the Hermite process of order 2 is the Rosenblatt process. We consider here the sum of two Hermite processes of orders and and of different Hurst parameters. We then study its quadratic variations at different scales. This is akin to a wavelet decomposition. We study both the cases where the Hermite processes are dependent and where they are independent. In the dependent case, we show that the quadratic variation, suitably normalized, converges either to a normal or to a Rosenblatt distribution, whatever the order of the original Hermite processes. [Copyright &y& Elsevier]

We consider stationary processes with long memory which are non-Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long–memory parameter. We show that the limit is not Gaussian but can be expressed using the non-Gaussian Rosenblatt process defined as a Wiener–Itô integral of order 2. This happens even if the original process is defined through a Hermite polynomial of order higher than 2. [ABSTRACT FROM AUTHOR]

We consider linear processes, not necessarily Gaussian, with long, short or negative memory. The memory parameter is estimated semi-parametrically using wavelets from a sample X1,..., X n of the process. We treat both the log-regression wavelet estimator and the wavelet Whittle estimator. We show that these estimators are asymptotically normal as the sample size n → ∞ and we obtain an explicit expression for the limit variance. These results are derived from a general result on the asymptotic normality of the scalogram for linear processes, conveniently centred and normalized. The scalogram is an array of quadratic forms of the observed sample, computed from the wavelet coefficients of this sample. In contrast to quadratic forms computed on the basis of Fourier coefficients such as the periodogram, the scalogram involves correlations which do not vanish as the sample size n → ∞. [ABSTRACT FROM AUTHOR]

In recent years, methods to estimate the memory parameter using wavelet analysis have gained popularity in many areas of science. Despite its widespread use, a rigorous semi-parametric asymptotic theory, comparable with the one developed for Fourier methods, is still lacking. In this article, we adapt to the wavelet setting, the classical semi-parametric framework introduced by Robinson and his co-authors for estimating the memory parameter of a (possibly) non-stationary process. Our results apply to a class of wavelets with bounded supports, which include but are not limited to Daubechies wavelets. We derive an explicit expression of the spectral density of the wavelet coefficients and show that it can be approximated, at large scales, by the spectral density of the continuous-time wavelet coefficients of fractional Brownian motion. We derive an explicit bound for the difference between the spectral densities. As an application, we obtain minimax upper bounds for the log-scale regression estimator of the memory parameter for a Gaussian process and we derive an explicit expression of its asymptotic variance. [ABSTRACT FROM AUTHOR]