Convergence of integrated processes of arbitrary Hermite rank

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 , is investigated. Here d(x) = x^HL(x) with <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 m1. (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 m1 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 m2. 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.Research supported by the National Science Foundation grants MCS 77-03543 and ENG 78-11454.This paper contains results closely connected to those of the paper by Dobrushin and Major, Z. Wahrscheinlichkeitstheorie verw. Gebiete 50, 27–52 (1979). The investigations were done independently and at about the same time. Different methods were usedDedicated to Professor Leopold Schmetterer on occasion of his 60th Birthday