Asymptotic behavior for an additive functional of two independent self-similar Gaussian processes

Abstract

We derive the asymptotic behavior for an additive functional of two independent self-similar Gaussian processes when their intersection local time exists, using the method of moments.(H2) Bounds on the variance of increments: There exist positive constants γ 0 ≥ 1, α 2 > 0 and nonnegative decreasing functionsfor all h ∈ [0, t/γ 0 ].(H3) Bounds on the covariance of increments on disjoint intervals: there exists a nonnegative decreasing function β(γ) : (1, ∞) → R with lim γ→∞ β(γ) = 0, such that, for any 0 < t 1 < t 2 < t 3 < t 4 < ∞ such that ∆t 2 ∆t 4 ≤ 1 γ or ∆t 2 ∆t 4 ≥ γ or max ∆t 2 ∆t 3 , ∆t 4 ∆t 3 ≤ 1 γ , then E X ℓ t 4 − X ℓ

Higher order derivative of self-intersection local time for fractional Brownian motion

“…But in this paper, we need the following nondeterminism property. By Nualart and Xu [9] (see also in Song, Xu and Yu [11]), we can see that for any n ∈ N, there exists two constants κ H and β H depending only on n and H, such that for any 0 = s…”

Derivatives of local times for some Gaussian fields

“…A reason for focusing on two independent Gaussian processes is that the methodology developed here can be easily used to obtain the corresponding results for one Gaussian process or k (k ≥ 3) independent Gaussian processes. Moreover, this paper can be viewed as an extension of [9,7] where central limit theorems for functionals of X H t − X H s are not available for H ≤ 2 d+2 , see [5] for this phenomenon in the one 1-dimensional fBm case. Here the main difficulty comes from the second independent Gaussian process.…”