Extreme value theory for moving average processes.

*(English)*Zbl 0604.60019This is an interesting qualitative and quantitative study of extreme values of moving averages of variables with smooth tails.

Let \(\{X_ t=\sum c_{\lambda -t}Z_{\lambda}\}\) be an infinite moving average process, with \(\{c_{\lambda}\}\) given constants and with the noise sequence \(\{Z_{\lambda}\}\) consisting of i.i.d. random variables. The author studies extremal properties connected with such processes, for the case when the marginal distribution of \(\{Z_{\lambda}\}\) has a tail which decreases approximately as a polynomial times \(\exp \{-z^ p\}\) as \(z\to \infty\), for \(0<p<\infty.\)

The main results of this paper concern convergence of point processes of heights and locations of extreme values of \(\{X_ t\}\) consisting of the points \((j/n,a_ n(X_ j-b_ n))\), \(j=1,2,...\), where \(a_ n>0\), \(b_ n\) are norming constants, and of more general ”marked” point processes which retain information also about the behaviour of sample paths near extremes. As corollaries the author obtains the convergence results for maxima.

In addition to extreme values of \(\{X_ t\}\) itself, the author studies their relation to extremes of the \(Z_{\lambda}'s\) and of an i.i.d. sequence \(\tilde X_ 1,\tilde X_ 2,..\). having the same marginal distribution function as the \(X_ t's\). In the last section, he also comments briefly on earlier results for polynomially decreasing tails.

Let \(\{X_ t=\sum c_{\lambda -t}Z_{\lambda}\}\) be an infinite moving average process, with \(\{c_{\lambda}\}\) given constants and with the noise sequence \(\{Z_{\lambda}\}\) consisting of i.i.d. random variables. The author studies extremal properties connected with such processes, for the case when the marginal distribution of \(\{Z_{\lambda}\}\) has a tail which decreases approximately as a polynomial times \(\exp \{-z^ p\}\) as \(z\to \infty\), for \(0<p<\infty.\)

The main results of this paper concern convergence of point processes of heights and locations of extreme values of \(\{X_ t\}\) consisting of the points \((j/n,a_ n(X_ j-b_ n))\), \(j=1,2,...\), where \(a_ n>0\), \(b_ n\) are norming constants, and of more general ”marked” point processes which retain information also about the behaviour of sample paths near extremes. As corollaries the author obtains the convergence results for maxima.

In addition to extreme values of \(\{X_ t\}\) itself, the author studies their relation to extremes of the \(Z_{\lambda}'s\) and of an i.i.d. sequence \(\tilde X_ 1,\tilde X_ 2,..\). having the same marginal distribution function as the \(X_ t's\). In the last section, he also comments briefly on earlier results for polynomially decreasing tails.

Reviewer: W.Dziubdziela

##### MSC:

60F05 | Central limit and other weak theorems |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

60G17 | Sample path properties |