6 research outputs found
Functional Limit Theorems for Toeplitz Quadratic Functionals of Continuous time Gaussian Stationary Processes
\noindent The paper establishes weak convergence in of normalized
stochastic processes, generated by Toeplitz type quadratic functionals of a
continuous time Gaussian stationary process, exhibiting long-range dependence.
Both central and non-central functional limit theorems are obtained
The Trace Problem for Toeplitz Matrices and Operators and its Impact in Probability
The trace approximation problem for Toeplitz matrices and its applications to
stationary processes dates back to the classic book by Grenander and Szeg\"o,
"Toeplitz forms and their applications". It has then been extensively studied
in the literature.
In this paper we provide a survey and unified treatment of the trace
approximation problem both for Toeplitz matrices and for operators and describe
applications to discrete- and continuous-time stationary processes.
The trace approximation problem serves indeed as a tool to study many
probabilistic and statistical topics for stationary models. These include
central and non-central limit theorems and large deviations of Toeplitz type
random quadratic functionals, parametric and nonparametric estimation,
prediction of the future value based on the observed past of the process, etc.
We review and summarize the known results concerning the trace approximation
problem, prove some new results, and provide a number of applications to
discrete- and continuous-time stationary time series models with various types
of memory structures, such as long memory, anti-persistent and short memory