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

Hu, Yaozhong; Nualart, David; Xu, Fangjun

doi:10.1214/12-aop825

Cited by 18 publications

(36 citation statements)

References 15 publications

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“…This theorem has been proved by Hu, Nualart and Xu in the reference [3], in the case where the Hurst parameter H satisfies 1 d+1 < H < 1 d , and it has been conjectured in that paper that the result can be extended to the case 1 d+2 < H ≤ 1 d+1 . The purpose of the present paper is to prove this conjecture.…”

Section: Introductionmentioning

confidence: 83%

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

Nualart

2013

Electron. Commun. Probab.

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Section: Introductionmentioning

confidence: 83%

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

Nualart

2013

Electron. Commun. Probab.

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“…This paper can be viewed as an extension of the result in [7]. The limit here is different from that in [7], which, conditionally on X (1) and X (2) , is a two-parameter Gaussian process. To prove our main result Theorem 1.1, we use Fourier analysis and the method of moments.…”

Section: Introductionmentioning

confidence: 91%

“…To prove our main result Theorem 1.1, we use Fourier analysis and the method of moments. 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.…”

Section: Introductionmentioning

confidence: 99%

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

Nualart

2019

Stochastic Processes and their Applications

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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

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“…Elimination of some negligible terms. The first term which might yield a negligible contribution in Q h is given by the small exponential term e −((t−r)ξ 2 )/2 in expression (9). We thus set Q h,β t,r = Q h,β,1…”

Section: 2mentioning

confidence: 99%

“…[7]) of Brownian local time. Malliavin calculus tools have also been essential in order to generalize the notion of self-intersection local time [5,8] and to obtain central limit theorems for additive functionals [9] of fractional Brownian motion.…”

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confidence: 99%

On L2 modulus of continuity of Brownian local times and Riesz potentials

Deya¹,

Nualart²,

Tindel³

2015

Ann. Probab.

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This article is concerned with modulus of continuity of Brownian local times. Specifically, we focus on three closely related problems: (a) Limit theorem for a Brownian modulus of continuity involving Riesz potentials, where the limit law is an intricate Gaussian mixture. (b) Central limit theorems for the projections of L 2 modulus of continuity for a one-dimensional Brownian motion. (c) Extension of the second result to a two-dimensional Brownian motion. Our proofs rely on a combination of stochastic calculus and Malliavin calculus tools, plus a thorough analysis of singular integrals.1. Introduction. Let {B t , 0 ≤ t ≤ 1} be a standard linear Brownian motion defined on some complete probability space (Ω, F, P). In the sequel, we denote by L t (x) the local time of B at a given point x ∈ R, defined for t ∈ [0, 1]. A nice combination of stochastic calculus, stochastic analysis and evaluation of singularities associated with heat kernels have recently led to a number of interesting limit theorems for quantities related to the family {L t (x); t ∈ [0, 1], x ∈ R}. Let us quote, for instance, the use of Malliavin and stochastic calculus tools in order to get suitably normalized limits for L 2 modulus of continuity (see [6,13]) or third moment in space (cf. [7]) of Brownian local time. Malliavin calculus tools have also been essential in order to generalize the notion of self-intersection local time [5,8] and to obtain central limit theorems for additive functionals [9] of fractional Brownian motion.The current article proposes to take another step into the relationships between Brownian local time and stochastic analysis. Specifically, we shall handle the following problems:

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

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

Cited by 18 publications

References 15 publications

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

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

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

On L2 modulus of continuity of Brownian local times and Riesz potentials

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