Extremes of Moving Averages of Random Variables with Finite Endpoint

Abstract

Consider moving average processes of the form $X_t = \sum^\infty_{j=0} c_jZ_{t-j}$, where $\{Z_j\}$ are iid and nonnegative random variables and $c_j > 0$ are constants satisfying summability conditions at least sufficient to make the random series above converge. We suppose that the distribution of $Z_j$ is regularly varying near 0 and discuss lower tail behavior of finite and infinite linear combinations. The behavior is quite different in the two cases. For finite linear combinations, the lower tail is again regularly varying but for infinite moving averages, the lower tail is $\Gamma$-varying, i.e., it is in the domain of attraction of a type I extreme value distribution in the sense of minima. Convergence of point processes based on the moving averages is shown to hold in both the finite and infinite order cases and suitable conclusions are drawn from such convergences. A useful analytic tool is asymptotic normality of the Esscher transform of the common distribution of the $Z$'s. The extreme value results of this paper are in terms of minima of the moving average processes but results can be adapted to study maxima of moving averages of random variables in the domain of attraction of the type III extreme value distribution for maxima.