Frontiers in Probability and the Statistical Sciences Random Toeplitz Functionals Their Applications

font-family: 'Calibri',sans-serif;">The book addresses topics that are often overlooked in other texts, including the trace approximation problem, central limit theorems in continuous time, functional central and non-central limit theorems for Toeplitz processes, and central limit theorems for tapered functionals. This book presents recent findings on central and non-central limit theorems for Toeplitz and tapered Toeplitz random quadratic functionals of stationary processes, with applications in spectral-based statistical inference. It focuses on Gaussian, orthogonal increment-driven, and Lévy-driven linear stationary processes with memory, in both discrete and continuous time. Toeplitz matrices and operators are central to the study of stationary processes. The covariance matrix of a discrete-time stationary process is a truncated Toeplitz matrix generated by the process's spectral density; in continuous-time, this becomes a Toeplitz operator. The foundations of the trace approximation problem were laid by Grenander and Szegö in their classical monograph “Toeplitz Forms and Their Applications” (1958), and the subject has recently seen renewed interest due to developments in long-range dependence and tapered data analysis. The book addresses topics that are often overlooked in other texts, including the trace approximation problem, central limit theorems in continuous time, functional central and non-central limit theorems for Toeplitz processes, and central limit theorems for tapered functionals. It also covers approaches to estimating linear and nonlinear spectral functionals, Whittle estimators, and goodness-of-fit tests using tapered data – each enriched by new advances in the field.