High order chaotic limits of wavelet scalograms under long-range dependence

Let G be a non–linear function of a Gaussian process {Xt}t∈Z with long–range dependence. The resulting process {G(Xt)}t∈Z is not Gaussian when G is not linear. We consider random wavelet coefficients associated with {G(Xt)}t∈Z and the corresponding wavelet scalogram which is the average of squares of wavelet coefficients over locations. We obtain the asymptotic behavior of the scalogram as the number of observations and the analyzing scale tend to infinity. It is known that when G is a Hermite polynomial of any order, then the limit is either the Gaussian or the Rosenblatt distribution, that is, the limit can be represented by a multiple Wiener-Itˆo integral of order one or two. We show, however, that there are large classes of functions G which yield a higher order Hermite distribution, that is, the limit can be represented by a a multiple Wiener-Itˆo integral of order greater than two. This happens for example if G is a linear combination of a Hermite polynomial of order 1 and a Hermite polynomial of order q > 3. The limit in this case can be Gaussian but it can also be a Hermite distribution of order q − 1 > 2. This depends not only on the relation between the number of observations and the scale size but also on whether q is larger or smaller than a new critical index q ∗ . The convergence of the wavelet scalogram is therefore significantly more complex than the usual one.M. Clausel's research was partially supported by the PEPS project AGREE and LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement d'avenir.F. Roueff's research was partially supported by the ANR project MATAIM NT09 441552Murad S.Taqqu was supported in part by the NSF grants DMS-1007616 at Boston University.C. Tudor's research was partially supported by the ANR grant Masterie BLAN 012103. (ANR-11-LABX-0025-01 - LabEx PERSYVAL-Lab - French program Investissement d'avenir; PEPS project AGREE; MATAIM NT09 441552 - ANR; BLAN 012103 - ANR; DMS-1007616 - NSF at Boston University)Accepted manuscrip