Central limit theorem for functionals of two independent fractional Brownian motions

Abstract

We prove a central limit theorem for functionals of two independent d-dimensional fractional Brownian motions with the same Hurst index H in ( 2 d+1 , 2 d ) using the method of moments.

Limit theorems for functionals of two independent Gaussian processes

“…When α 1 = α 2 , for example in the fBm case, λ = 1 and it is easy to see Z 1 = 1 a.s., and this is consistent with the known results in [2,11,12,16]. When α 1 = α 2 , for example in the sub-fBm or bi-fBm case, λ = 1 and Z λ is a non-trivial random variable.…”

“…(C1) For any 0 < t 1 < t 2 < t 3 < t 4 < ∞ and γ > 1, there exists a nonnegative decreasing function β 1 (γ) with lim γ→∞ β 1 (γ) = 0 such that, if ∆t 2 ∆t 4 ≤ 1 γ or ∆t 2 ∆t 4 ≥ γ, then E X 1 Remark 1.2 Note that the stationary increment property was used to obtain the limit laws for functionals of fBm or fBms in the previous literatures [2,11,12,16]. In this work, we do not require the stationary increment property, but instead assume some weaker conditions (A1) and (A2).…”

“…The other is that the role played by the second fBm in the limit laws is not well understood. Recently, Nualart and Xu in [12] proved central limit theorems for functionals of two independent d-dimensional fractional Brownian motions in the case Hd < 2. After that, Bi and Xu in [2] showed the first-order limit law in the critical case Hd = 2 with H ≤ 1/2, but it does not include the interesting case d = 3 which may have physical relevance.…”

“…Moreover, the second Gaussian process causes a big trouble when proving the convergence of even moments. Thanks to the methodologies developed in the recent papers [12,2,16] and the introduction of some new ideas, especially the pairing technique, these issues are solved eventually.…”

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

“…The next theorem is the main result of this paper, which is a functional version of the central limit theorem in the case where X = B H is a d-dimensional fractional Brownian motion (fBm) with Hurst parameter H, proved in [11].…”