The Annals of Statistics

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

Yuzo Hosoya

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This paper provides limit theorems for multivariate, possibly non-Gaussian 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 Fejér 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.

Article information

Ann. Statist. Volume 25, Number 1 (1997), 105-137.

First available in Project Euclid: 10 October 2002

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Mathematical Reviews number (MathSciNet)

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Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62M15: Spectral analysis 62E30
Secondary: 60G10: Stationary processes 60F05: Central limit and other weak theorems

Asymptotic theory central limit theorems mixing conditions martingale difference serial covariances quadratic forms bracketing function long-range dependence maximum likelihood estimation likelihood ratio test


Hosoya, Yuzo. A limit theory for long-range dependence and statistical inference on related models. Ann. Statist. 25 (1997), no. 1, 105--137. doi:10.1214/aos/1034276623.

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