Hermite ranks and $$U$$ U -statistics

C. Lévy-Leduc () and M. Taqqu ()

Metrika: International Journal for Theoretical and Applied Statistics, 2014, vol. 77, issue 1, 105-136

Abstract: We focus on the asymptotic behavior of $$U$$ U -statistics of the type $$\begin{aligned} \sum _{1\le i\ne j\le n} h(X_i,X_j)\\ \end{aligned}$$ ∑ 1 ≤ i ≠ j ≤ n h ( X i , X j ) in the long-range dependence setting, where $$(X_i)_{i\ge 1}$$ ( X i ) i ≥ 1 is a stationary mean-zero Gaussian process. Since $$(X_i)_{i\ge 1}$$ ( X i ) i ≥ 1 is Gaussian, $$h$$ h can be decomposed in Hermite polynomials. The goal of this paper is to compare the different notions of Hermite rank and to provide conditions for the remainder term in the decomposition to be asymptotically negligeable. Copyright Springer-Verlag Berlin Heidelberg 2014

Keywords: Long-range dependence; Hermite polynomials; $$U$$ U -statistics; 60G18; 62M10 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s00184-013-0474-4

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