Asymptotic expansions for functions of the increments of certain Gaussian processes

Let G={G(x),x≥0}G={G(x),x≥0} be a mean zero Gaussian process with stationary increments and set σ2(|x−y|)=E(G(x)−G(y))2σ2(|x−y|)=E(G(x)−G(y))2. Let ff be a function with Ef2(η)<∞Ef2(η)<∞, where η=N(0,1)η=N(0,1). When σ2σ2 is regularly varying at zero and limh→0h2σ2(h)=0andlimh→0σ2(h)h=0but (d2ds2σ2(s))j0 is locally integrable for some integer j0≥1j0≥1, and satisfies some additional regularity conditions, ∫abf(G(x+h)−G(x)σ(h))dx=∑j=0j0(h/σ(h))jE(Hj(η)f(η))j!:(G′)j:(I[a,b])+o(hσ(h))j0 in L2L2. Here HjHj is the jjth Hermite polynomial. Also :(G′)j:(I[a,b]):(G′)j:(I[a,b]) is a jjth order Wick power Gaussian chaos constructed from the Gaussian field G′(g)G′(g), with covariance E(G′(g)G′(g˜))=∬ρ(x−y)g(x)g˜(y)dxdy, where ρ(s)=12d2ds2σ2(s).