Abstract
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.
Highlights
A significant part of large-sample statistical inference relies on limit theorems of probability theory, which involves sums and quadratic functionals of stationary observations
We present results on central and non-central limit theorems for Toeplitz and tapered Toeplitz type quadratic functionals of stationary processes with applications in parametric and nonparametric statistical estimation theory
Central and non-central limit theorems for tapered quadratic forms of a d.t. long memory Gaussian stationary fields have been proved in Doukhan et al [32]
Summary
A significant part of large-sample statistical inference relies on limit theorems of probability theory, which involves sums and quadratic functionals of stationary observations. The term ‘central limit theorem’ (CLT) refers to a statement that a suitably standardized quadratic functional converges in distribution to a Gaussian random variable. Limit theorems where a suitably standardized quadratic functional converges in distribution to a nonGaussian random variable are termed ‘non-central limit theorems’ (NCLT). We present results on central and non-central limit theorems for Toeplitz and tapered Toeplitz type quadratic functionals of stationary processes with applications in parametric and nonparametric statistical estimation theory. We discuss some questions concerning Toeplitz matrices and operators, Fejer-type singular integrals, Levy-Ito-type and Stratonovich-type multiple stochastic integrals, and power counting theorems. These are the main tools for obtaining limit theorems, but they are of interest in themselves
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