Hermite ranks and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$U$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>U</mi> </math> </EquationSource> </InlineEquation>-statistics

We focus on the asymptotic behavior of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$U$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>U</mi> </math> </EquationSource> </InlineEquation>-statistics of the type <Equation ID="Equ73"> <EquationSource Format="TEX">$$\begin{aligned} \sum _{1\le i\ne j\le n} h(X_i,X_j)\\ \end{aligned}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink" display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munder> <mo>∑</mo> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≠</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>X</mi> <mi>j</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math> </EquationSource> </Equation>in the long-range dependence setting, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$(X_i)_{i\ge 1}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>i</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </math> </EquationSource> </InlineEquation> is a stationary mean-zero Gaussian process. Since <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$(X_i)_{i\ge 1}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>i</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </math> </EquationSource> </InlineEquation> is Gaussian, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$h$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>h</mi> </math> </EquationSource> </InlineEquation> 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