CLT for L^{p} moduli of continuity of Gaussian processes

Abstract

Let G=\{G(x),x\in R^1\} be a mean zero Gaussian processes with stationary increments and set \si ^2(|x-y|)= E(G(x)-G(y))^2. Let f be a symmetric function with Ef(\eta)<\ff, where \eta=N(0,1). When \si^2(s) is concave or when \si^2(s)=s^r$, $1<r\leq 3/2 we have \lim_{h\downarrow 0}{\int_a^b f(\frac{G(x+h)-G(x)}{\si (h)}) dx - (b-a)Ef(\eta)\over \sqrt{\Phi(h,\si(h),f,a,b)}}= N(0,1) in law where \Phi(h,\si(h),f,a,b) is the variance of the numerator. This result continues to hold when \si^2(s)=s^r, 3/2<r<2, for certain functions f, depending on the nature of the coefficients in their Hermite polynomial expansion. The asymptotic behavior of \Phi(h,\si(h),f,a,b) at zero, is described in a very large number of cases.