On a Szegö type limit theorem, the Hölder-Young-Brascamp-Lieb inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields

Abstract

Abstract. Many statistical applications require establishing central limit theorems for sums/integrals Mathematics Subject Classification. 60F05, 62M10, 60G15, 62M15, 60G10, 60G60.

On the robustness to small trends of parameter estimation for continuous-time stationary models with memory

“…It can be shown that the standard Taylor expansion methods based on the smoothed periodogram convergence results of type (3.4) and (3.5) with a general smoothing function g(λ; θ) and with the contaminated periodogram I T,X (λ) instead of I T,Y (λ), lead consistent and asymptotically normally distributed estimators of θ. We will not pursue this matter here (the details will be reported elsewhere), however, notice that in the special case of Whittle procedure, where g(λ; θ) = ∂ ∂ 1 f (λ,θ) · w(λ) the results of Anh et al [3], Avram et al [5], Casas and Gao [9], Gao [14], Gao et al [15,16], Leonenko and Sakhno [26] concerning consistency and asymptotic normality of the Whittle minimum contrast estimators constructed on the basis of the periodogram I T,Y (λ), continue to hold without change for estimators calculated on the basis of the contaminated periodogram I T,X (λ), under appropriate assumptions imposed on the model Y (t) on the smoothing function g(λ, θ) and on the trend M (t).…”

“…Continuous versions of Whittle estimation procedure have been considered, for example, in Anh et al [3,4], Avram et al [5], Casas and Gao [9], Gao [14], Gao et al [15,16], Leonenko and Sakhno [26].…”

“…and Using this approach, statistical properties of Whittle minimum contrast estimators for continuous-time stationary processes were studied in Anh et al [3], Avram et al [5], Casas and Gao [9], Gao [14], Gao et al [15,16], Leonenko and Sakhno [26]. In particular, consistency and asymptotic normality of Whittle minimum contrast estimatorθ W was established for some classes of stationary models, including the fractional Riesz-Bessel motion model, specified by spectral density f (λ) = f (λ; θ) given by (2.4) with θ = (u, v, c).…”

“…is a central limit theorem for Toeplitz type quadratic functionals of stationary processes (see Ginovyan [18,19], Ginovyan and Sahakyan [21] for Gaussian processes, and Bai et al [5,6] for linear processes). It can be shown that the standard Taylor expansion methods based on the smoothed periodogram convergence results of type (3.4) and (3.5) with a general smoothing function g(λ; θ) and with the contaminated periodogram I T,X (λ) instead of I T,Y (λ), lead consistent and asymptotically normally distributed estimators of θ.…”

“…For instance, in Anh et al [3,4], Avram et al [5], Casas and Gao [9], Gao [14], Gao et al [15,16], Leonenko and Sakhno [26] were obtained sufficient conditions ensuring consistency and asymptotic normality of various statistical estimators of θ, including quasi maximum likelihood (Whittle) and minimum contrast estimators of θ, constructed on the basis of a finite realization Y T := {Y (t), 0 ≤ t ≤ T } of the process Y (t).…”