Asymptotic expansions for functions of the increments of certain Gaussian processes

Let G = { G ( x ) , x ≥ 0 } 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 function with E f 2 ( η ) < ∞ , where η = N ( 0 , 1 ) . When σ 2 is regularly varying at zero and lim h → 0 h 2 σ 2 ( h ) = 0 and lim h → 0 σ 2 ( h ) h = 0 but  ( d 2 d s 2 σ 2 ( s ) ) j 0 is locally integrable for some integer j 0 ≥ 1 , and satisfies some additional regularity conditions, ∫ a b f ( G ( x + h ) − G ( x ) σ ( h ) ) d x = ∑ j = 0 j 0 ( h / σ ( h ) ) j E ( H j ( η ) f ( η ) ) j ! : ( G ′ ) j : ( I [ a , b ] ) + o ( h σ ( h ) ) j 0 in L 2 . Here H j is the j th Hermite polynomial. Also : ( G ′ ) j : ( I [ a , b ] ) is a j th order Wick power Gaussian chaos constructed from the Gaussian field G ′ ( g ) , with covariance E ( G ′ ( g ) G ′ ( g ˜ ) ) = ∬ ρ ( x − y ) g ( x ) g ˜ ( y ) d x d y , where ρ ( s ) = 1 2 d 2 d s 2 σ 2 ( s ) .