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.