High order chaotic limits of wavelet scalograms under long-range dependence
Files
Accepted manuscript
Date
2013-01-01
DOI
Authors
Clausel, M.
Roueff, F.
Taqqu, Murad S.
Tudor, C.
Version
Accepted manuscript
OA Version
Citation
M Clausel, F Roueff, MS Taqqu, C Tudor. 2013. "High order chaotic limits of wavelet scalograms under long-range dependence." ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS, Volume 10, Issue 2, pp. 979 - 1011 (33).
Abstract
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.