Central limit theorem for an additive functional of the fractional Brownian motion II

Abstract

We prove a central limit theorem for an additive functional of the d-dimensional fractional Brownian motion with Hurst index H ∈ ( 1 2+d , 1 d ), using the method of moments, extending the result by Papanicolaou, Stroock and Varadhan in the case of the standard Brownian motion.

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.…”

“…(II) Chaining argument. In this part, we apply the chaining argument introduced in [11] to the integral on the right-hand side of (4.8). The main idea is to replace each product f (y 2i−1 − y 2i ) f (y 2i − y 2i+1 ) by f (−y 2i ) f (y 2i ) = | f (y 2i )| 2 , noting that, by the assumption R d |f (x)||x| β dx < ∞ for some β > 0, the differences f (y 2i−1 − y 2i ) − f (−y 2i ) and f (y 2i − y 2i+1 ) − f (y 2i ) are bounded by constant multiples of |y 2i−1 | α and |y 2i+1 | α , respectively, for any α ∈ [0, β].…”

“…(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).…”

Second-order limit laws for occupation times of fractional Brownian motion

“…To show Theorems 1.1 and 1.2, we would use the method of moments plus some kind of chaining argument. The chaining argument was first developed in [7] to prove the central limit theorem for an additive functional of the d-dimensional fractional Brownian motion with Hurst index H ∈ ( 1 d+2 , 1 d ). It has been proved to be very powerful when obtaining the asymptotic behavior of moments.…”

“…It has been proved to be very powerful when obtaining the asymptotic behavior of moments. However, the situation here is a little different from that in [7]. We consider fractional Brownian motions in the critical case and use a different normalizing factor.…”