Functional Limit Theorems for Toeplitz Quadratic Functionals of Continuous time Gaussian Stationary Processes (original) (raw)
Related papers
2021
This is a survey of recent results on central and non-central limit theorems for quadratic functionals of stationary processes. The underlying processes are Gaussian, linear or Lévy-driven linear processes with memory, and are defined either in discrete or continuous time. We focus on limit theorems for Toeplitz and tapered Toeplitz type quadratic functionals of stationary processes with applications in parametric and nonparametric statistical estimation theory. We discuss questions concerning Toeplitz matrices and operators, Fejér-type singular integrals, and Lévy-Itô-type and Stratonovich-type multiple stochastic integrals. These are the main tools for obtaining limit theorems.
Limit theorems for Toeplitz-type quadratic functionals of stationary processes and applications
Probability Surveys, 2022
This is a survey of recent results on central and non-central limit theorems for quadratic functionals of stationary processes. The underlying processes are Gaussian, linear or Lévy-driven linear processes with memory, and are defined either in discrete or continuous time. We focus on limit theorems for Toeplitz and tapered Toeplitz type quadratic functionals of stationary processes with applications in parametric and nonparametric statistical estimation theory. We discuss questions concerning Toeplitz matrices and operators, Fejér-type singular integrals, and Lévy-Itô-type and Stratonovich-type multiple stochastic integrals. These are the main tools for obtaining limit theorems.
2019
Let {X(t), t ∈ ℝ} be a centered real-valued stationary Gaussian process with spectral density f. The paper considers a question concerning asymptotic distribution of tapered Toeplitz type quadratic functional \(Q_T^h\) of the process X(t), generated by an integrable even function g and a taper function h. Sufficient conditions in terms of functions f, g and h ensuring central limit theorems for standard normalized quadratic functionals \(Q_T^h\) are obtained, extending the results of Ginovyan and Sahakyan (Probability Theory and Related Fields 138, 551–579, 2007) to the tapered case and sharpening the results of Ginovyan and Sahakyan (Electronic Journal of Statistics 13, 255–283, 2019) for the Gaussian case.
CLT and other limit theorems for functionals of Gaussian processes
Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 1985
Conditions for the CLT for non-linear functionals of stationary Gaussian sequences are discussed, with special references to the borderline between the CLT and the non-CLT. Examples of the non-CLT for such functionals with the norming factor t/N are given.
Stochastic Processes and their Applications, 1989
Let X=(X,, t CR) be a stationary Gaussian process on (R, .F, P) with time-shift operators ( CJ,, s E R) and let H(X) = L'(f), u(X), P) denote the space of square-integrable functionals of X. Say that Y t H(X) with EY =O satisfies the Central Limit Theorem (CLT) if A family of martingales (Z,(t), t 2 0) is exhibited for which 2, ("c) = Z,, and martingale techniques and results are used to provide suficient conditions on X and Y for the CLT. These conditions are then shown to be necessary for slightly more restrictive central limit behavior of Y.
The Annals of Probability, 2013
The limit Gaussian distribution of multivariate weighted functionals of nonlinear transformations of Gaussian stationary processes, having multiple singular spectra, is derived, under very general conditions on the weight function. This paper is motivated by its potential applications in nonlinear regression, and asymptotic inference on nonlinear functionals of Gaussian stationary processes with singular spectra.
On the Central Limit Theorem for Toeplitz Quadratic Forms
2005
Let X(t), t = 0, ±1,. .. , be a real-valued stationary Gaussian sequence with a spectral density function f (λ). The paper considers the question of applicability of the central limit theorem (CLT) for a Toeplitz-type quadratic form Qn in variables X(t), generated by an integrable even function g(λ). Assuming that f (λ) and g(λ) are regularly varying at λ = 0 of orders α and β, respectively, we prove the CLT for the standard normalized quadratic form Qn in a critical case α + β = 1 2. We also show that the CLT is not valid under the single condition that the asymptotic variance of Qn is separated from zero and infinity.
A functional limit theorem for ´-weakly dependent processes and its applications
2006
We prove a general functional central limit theorem for weak dependent time series. A very large variety of models, for instance, causal or non causal linear, ARCH(∞), bilinear, Volterra processes, satisfies this theorem. Moreover, it provides numerous application as well for bounding the distance between the empirical mean and the Gaussian measure than for obtaining central limit theorem for sample moments and cumulants.