Limit theorems for functionals of two independent Gaussian processes

Abstract

Under certain mild conditions, some limit theorems for functionals of two independent Gaussian processes are obtained. The results apply to general Gaussian processes including fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. A new and interesting phenomenon is that, in comparison with the results for fractional Brownian motion, extra randomness appears in the limiting distributions for Gaussian processes with nonstationary increments, say sub-fractional Brownian motion and bi-fractional Brownian. The results are obtained based on the method of moments, in which Fourier analysis, the chaining argument introduced in [11] and a paring technique are employed.

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

“…The estimates obtained by using the chaining argument play a critical role when deriving limit theorems for an additive functional of the fractional Brownian motion (fBm), see [10,17]. In fact, using some estimates on the covariance of increments of the fBm on disjoint intervals (see Lemma 2.4 in [14]), the convergence of even moments in [7,10] can be easily obtained applying the method given in the proof of Proposition 4.2 of [17]. However, these estimates could not help us to obtain the central limit theorems for an additive functional of two independent fBms, or more generally of two independent copies of a Gaussian process X satisfying conditions (H1)-(H3), because the methodology developed in [11] only allows us to derive obtain central limit theorems when times are fixed.…”

“…A new technique will be introduced to extend the result in [11] to functional central limit theorems, which is called the paring technique and was original developed in [14] to get limit laws for functionals of two independent fBms in the critical case Hd = 2. Here is a rough description of this technique.…”

“…More interestingly, the paring for the first process would also match the paring for the second one. These parings finally let us obtain the desired convergence of even moments in [14]. In this paper, we consider the case Hd < 2, where the intersection local exists, and the approach will be different from the one in [14] and new ideas and tools will be required to derive the functional central limit theorem.…”

“…Some techniques in [7] will be applied, but new ideas are also needed. For example, we use the paring technique introduced in [14] to prove the convergence of moments. On the other hand, this paper sheds light on proving asymptotic behavior of additive functionals of multi-parameter processes or random fields when intersection local time or local time exists.…”

“…A first-order result for two independent fBms in the critical case Hd = 2 with H ≤ 1/2 was given in [2]. Recently, both first-order and second-order results for two independent general Gaussian processes in the critical case Hd = 2 were proved in [14], where the functional limit theorems are still unknown. Extensions to functionals of k independent Gaussian processes in the critical case Hd = k are also mentioned in [14].…”