# Convergence of integrated processes of arbitrary Hermite rank

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## 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 that*Z*_{ 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## Preview

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