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 scales tends 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-Itô 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-Itô 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. Date: January 23, 2012. 2010 Mathematics Subject Classification. Primary 42C40 ; 60G18 ; 62M15 ; Secondary: 60G20, 60G22.