Rate of Convergence for the Weighted Hermite Variations of the Fractional Brownian Motion
- PDF / 440,810 Bytes
- 29 Pages / 439.37 x 666.142 pts Page_size
- 41 Downloads / 160 Views
Rate of Convergence for the Weighted Hermite Variations of the Fractional Brownian Motion Nicholas Ma1
· David Nualart1
Received: 7 November 2018 / Revised: 18 June 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this paper, we obtain a rate of convergence in the central limit theorem for high order weighted Hermite variations of the fractional Brownian motion. The proof is based on the techniques of Malliavin calculus and the quantitative stable limit theorems proved by Nourdin et al. (Ann Probab 44:1–41, 2016). Keywords Weighted Hermite variations · Malliavin calculus · Fractional Brownian motion Mathematics Subject Classification (2010) 60F05 · 60H07 · 60G22
1 Introduction The fractional Brownian motion B = {Bt , t ≥ 0} is characterized by being a zeromean Gaussian self-similar process with stationary increments and variance E(Bt2 ) = t 2H . The self-similarity index H ∈ (0, 1) is called the Hurst parameter. The fractional Brownian motion was first introduced by Kolmogorov in 1940. However, the landmark paper by Mandelbrot and Van Ness [7] gave fractional Brownian motion its name and inspired much of the modern literature on the subject. The study of single path behavior of stochastic processes often uses their power variations. In particular, the fractional Brownian motion is known to have a 1/H variation on any finite time interval equal to the length of the interval multiplied by the constant κ H = E[|Z 1/H |], where Z is a N (0, 1) random variable. That means, if
The work of D. Nualart is supported by the NSF Grant DMS 1811181.
B
Nicholas Ma [email protected] David Nualart [email protected]
1
Department of Mathematics, The University of Kansas, Lawrence, Kansas 66045, USA
123
Journal of Theoretical Probability
we consider the uniform partition of the interval [0, 1] into n ≥ 1 intervals and for 0 ≤ k ≤ n − 1 we denote Bk/n = B(k+1)/n − Bk/n , we have lim
n−1
n→∞
|Bk/n |1/H → κ H ,
k=0
where the convergence holds almost surely and in L p () for any p ≥ 2. A central limit theorem associated with this approximation can be obtained by expanding the function |x|1/H into Hermite polynomials. In particular, as a consequence of the Breuer–Major 1 , we have the convergence theorem [5], for each integer q ≥ 2 such that H < 1 − 2q in law n−1 1 L lim √ Hq (n H Bk/n ) → N (0, σ H2 ,q ), n→∞ n k=0
where Hq is the qth Hermite polynomial and σ H2 ,q = q!
ρ H (k)q .
(1.1)
k∈Z
1 (|k + 1|2H + |k − 1|2H − 2|k|2H ), k∈Z (1.2) 2 denotes the covariance of the stationary sequence {Bk+1 − Bk , k ≥ 0}. There has been intensive research on the asymptotic behavior of the weighted Hermite variations of the fractional Brownian motion B, defined by Here
ρ H (k) =
n−1 1 f (Bk/n )Hq (n H Bk/n ), Fn = √ n
(1.3)
k=0
where f is a given function. The analysis of the asymptotic behavior of these quantities is motivated, for instance, by the study of the exact rates of convergence of some approximation schemes of scalar stochastic differential equations driven by the fractiona
Data Loading...
