CONVERGENCE OF INTEGRATED PROCESSES OF ARBITRARY HERMITE RANK

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 {Mathematical expression}, is investigated. Here d(x) = xHL(x) with {Mathematical expression}<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 that Zx(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. © 1979 Springer-Verlag.