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Convergence of integrated processes of arbitrary Hermite rank

  • Murad S. Taqqu
Article

Summary

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) = x H L(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, EG 2 (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 thatZ x (t) converges for all m≧1 to some process ¯Z m (t) that depends essentially on m. The limiting process ¯Z m (t) is characterized through various representations involving multiple Itô integrals. These representations are all equivalent in the finite-dimensional distributions sense. The processes ¯Z m (t) are non-Gaussian when m≧2. They are self-similar, that is,¯Z m (at) and a H ¯Z m (t) have the same finite-dimensional distributions for all a>0.

Keywords

Stochastic Process Probability Theory Mathematical Biology Gaussian Process Integrate Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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