Estimating the Hurst Parameter

Abstract

60F05, 60G15, 60G18, 62F12, 33C45, central limit theorem, estimation, fractional Brownian motion, Gaussian processes, Hermite polynomials,

Quantitative Breuer–Major theorems

“…Note that any central limit result involving Hermite ranks and series of covariance coefficients is customarily called a 'Breuer-Major theorem', in honor of the seminal paper [8]. Theorem 1.1 and its variations have served as fundamental tools for Gaussian approximations in an impressive number of applications, of which we provide a representative (recent) sample: renormalization of fractional diffusions [1], power variations of Gaussian and Gaussian-related continuous-time processes [3,4,14,20,24], Gaussian fluctuations of heat-type equations [5], estimation of Hurst parameters of fractional processes [9,12,13], unit-root problems in econometrics [10], empirical processes of long-memory time series [22], level functionals of stationary Gaussian fields [19], variations of multifractal random walks [21], and stochastic programming [38]. See also Surgailis [34] for a survey of some earlier uses of Breuer-Major criteria.…”

Estimation in models driven by fractional Brownian motion

“…σ(·) ≡ 1 and µ(·) ≡ 0 and the purpose was to estimate H. Indeed, we prove that the asymptotic behavior of such estimators, that is, ( H k − H)/ √ ε and ( σ k − σ)/( √ ε log(ε)) are both equivalent to those of certain non-linear functionals of the Gaussian process b ε H (·) = ϕ ε * b H (·). As in [3], we show that they satisfy a central limit theorem using the method of moments, via the diagram formula. It is interesting to note that the rates of convergence of such estimators are not the same.…”

“…Theorem 2.2 gives the required result (the computation of the coefficients in the Hermite expansion of function g k (·) is explicitly made in the proof of Corollary 3.2 of [3]).…”

“…The symbol "⇒" will mean weak convergence of measures. At this step of the paper it will be helpful to state several theorems obtained in [3] in the aim to enlighten the notations and to make this paper more independent of it.…”

Quantitative Breuer-Major Theorems

“…Theorem 1.1 and its variations have served as fundamental tools for Gaussian approximations in an impressive number of applications, of which we provide a representative (recent) sample: renormalization of fractional diffusions [1], power variations of Gaussian and Gaussian-related continuous-time processes [3,4,14,20,24], Gaussian fluctuations of heat-type equations [5], estimation of Hurst parameters of fractional processes [9,12,13], unit-root problems in econometrics [10], empirical processes of long-memory time series [22], level functionals of stationary Gaussian fields [19], variations of multifractal random walks [21], and stochastic programming [38]. See also Surgailis [34] for a survey of some earlier uses of Breuer-Major criteria.…”