Convergence of integrated processes of arbitrary Hermite rank

  • Murad S. Taqqu

DOI: 10.1007/BF00535674

Cite this article as:
Taqqu, M.S. Z. Wahrscheinlichkeitstheorie verw Gebiete (1979) 50: 53. doi:10.1007/BF00535674


Let {X(s), −∞<s<∞} be a normalized stationary Gaussian process with a long-range correlation. The weak limit in C[0,1] of the integrated process \(Z_x \left( t \right) = \frac{1}{{d\left( x \right)}}\mathop \smallint \limits_0^{xt} G\left( {X\left( s \right)} \right)ds,{\text{ }}x \to \infty\), is investigated. Here d(x) = xHL(x) with \(\frac{1}{2}\)<H<1 and L(x) is a slowly varying function at infinity. The function G satisfies EG(X(s))=0, EG2(X(s))<∞ and has arbitrary Hermite rank m≧1. (The Hermite rank of G is the index of the first non-zero coefficient in the expansion of G in Hermite polynomials.) It is shown thatZx(t) converges for all m≧1 to some process ¯Zm(t) that depends essentially on m. The limiting process ¯Zm(t) is characterized through various representations involving multiple Itô integrals. These representations are all equivalent in the finite-dimensional distributions sense. The processes ¯Zm(t) are non-Gaussian when m≧2. They are self-similar, that is,¯Zm(at) and aH¯Zm(t) have the same finite-dimensional distributions for all a>0.

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Murad S. Taqqu
    • 1
  1. 1.School of Operations ResearchCornell UniversityIthaca, New YorkUSA

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