CLT for moduli of continuity of Gaussian processes

Let G = { G ( x ) , x ∈ R 1 } be a mean zero Gaussian process with stationary increments and set σ 2 ( | x − y | ) = E ( G ( x ) − G ( y ) ) 2 . Let f be a symmetric function with E f 2 ( η ) < ∞ , where η = N ( 0 , 1 ) . When σ 2 ( s ) is concave or when σ 2 ( s ) = s r , 1 < r ≤ 3 / 2 , lim h ↓ 0 ∫ a b f ( G ( x + h ) − G ( x ) σ ( h ) ) d x − ( b − a ) E f ( η ) Φ ( h , σ ( h ) , f , a , b ) = law N ( 0 , 1 ) where Φ ( h , σ ( h ) , f , a , b ) is the variance of the numerator. This result continues to hold when σ 2 ( s ) = s r , 3 / 2 < r < 2 , for certain functions f , depending on the nature of the coefficients in their Hermite polynomial expansion. ymptotic behavior of Φ ( h , σ ( h ) , f , a , b ) at zero is described in a very large number of cases.