The Annals of Statistics

The sample autocorrelations of heavy-tailed processes with applications to ARCH

Richard A. Davis and Thomas Mikosch

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We study the sample ACVF and ACF of a general stationary sequence under a weak mixing condition and in the case that the marginal distributions are regularly varying. This includes linear and bilinear processes with regularly varying noise and ARCH processes, their squares and absolute values. We show that the distributional limits of the sample ACF can be random, provided that the variance of the marginal distribution is infinite and the process is nonlinear. This is in contrast to infinite variance linear processes. If the process has a finite second but infinite fourth moment, then the sample ACF is consistent with scaling rates that grow at a slower rate than the standard $\sqrt{n}$. Consequently, asymptotic confidence bands are wider than those constructed in the classical theory. We demonstrate the theory in full detail for an ARCH (1) process.

Article information

Ann. Statist., Volume 26, Number 5 (1998), 2049-2080.

First available in Project Euclid: 21 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62G20 60G55 62P05 60G10 60G70: Extreme value theory; extremal processes

Point process vague convergence multivariate regular variation mixing condition stationary process heavy tail sample autocovariance sample autocorrelation ARCH finance Markov chain


Davis, Richard A.; Mikosch, Thomas. The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26 (1998), no. 5, 2049--2080. doi:10.1214/aos/1024691368.

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