A limit theory for long-range dependence and statistical inference on related models

This paper provides limit theorems for multivariate, possibly nonGaussian stationary processes whose spectral density matrices may have singularities not restricted at the origin, applying those limiting results to the asymptotic theory of parameter estimation and testing for statistical models of long-range dependent processes. The central limit theorems are proved based on the assumption that the innovations of the stationary processes satisfy certain mixing conditions for their conditional moments, and the usual assumptions of exact martingale difference or the (transformed) Gaussianity for the innovation process are dispensed with. For the proofs of convergence of the covariances of quadratic forms, the concept of the multiple Fejer kernel is introduced. For the derivation of the asymptotic properties of the quasi-likelihood estimate and the quasi-likelihood ratio, the bracketing function approach is used instead of conventional regularity conditions on the model spectral density. 0. Introduction. This paper investigates vector-valued stationary processes with long-range dependence which possess a variety of singularities not necessarily limited to zero frequency, giving a limit theory for quadratic forms of observations from those processes. Then it considers the statistical inference based on the quasi-likelihood function giving the asymptotic properties for the quasi-maximum likelihood (QML) estimate and the quasi-likelihood ratio (QLR) statistics based on the limit theory under very general conditions. The asymptotic results obtained reveal a particular feature of long-range dependence whose modeling produces different asymptotics for related statistics based on the quasi-likelihood function. A general framework for the asymptotic theory for parameter estimation and testing for short-range dependent stationary time-series models was given by Whittle (1952). Dealing with a linear scalar-valued process {zt, t E J} given by

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