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Consider a non--linear function $G(X_t)$ where $X_t$ is a stationary Gaussiansequence with long--range dependence. The usual reduction principle states thatthe partial sums of $G(X_t)$ behave asymptotically like the partial sums of thefirst term in the expansion of $G$ in Hermite polynomials. In the context ofthe wavelet estimation of the long--range dependence parameter, one replacesthe partial sums of $G(X_t)$ by the wavelet scalogram, namely the partial sumof squares of the wavelet coefficients. Is there a reduction principle in thewavelet setting, namely is the asymptotic behavior of the scalogram for$G(X_t)$ the same as that for the first term in the expansion of $G$ in Hermitepolynomial? The answer is negative in general. This paper provides a minimalgrowth condition on the scales of the wavelet coefficients which ensures thatthe reduction principle also holds for the scalogram. The results are appliedto testing the hypothesis that the long-range dependence parameter takes aspecific value.

Large scale reduction principle and application to hypothesis testing