The Annals of Applied Probability

Extreme value theory for a class of nonstationary time series with applications

Xu-Feng Niu

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Consider a class of nonstationary time series with the form $Y_t = \mu_t + \xi_t$ where ${\xi_t}$ is a sequence of infinite moving averages of independent random variables with regularly varying tail probabilities and different scale parameters. In this article, the extreme value theory of ${Y_t}$ is studied. Under mild conditions, convergence results for a point process based on the moving averages are proved, and extremal properties of the nonstationary time series, including the convergence of maxima to extremal processes and the limit point process of exceedances, are derived. The results are applied to the analysis of tropospheric ozone data in the Chicago area. Probabilities of monthly maximum ozone concentrations exceeding some specific levels are estimated, and the mean rate of exceedances of daily maximum ozone over the national standard 120 ppb is also assessed.

Article information

Ann. Appl. Probab., Volume 7, Number 2 (1997), 508-522.

First available in Project Euclid: 14 October 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Extreme value distributions Poisson random measure tropospheric ozone analysis threshold exceedances


Niu, Xu-Feng. Extreme value theory for a class of nonstationary time series with applications. Ann. Appl. Probab. 7 (1997), no. 2, 508--522. doi:10.1214/aoap/1034625342.

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