Extremes of Volterra Series Expansions with Heavy-Tailed Innovations

Linear time series models have widely been used in many areas of science. However, it is becoming increasingly clear that linear models do not represent adequately features often exhibited by data sets from environmental and economical processes. One property which is important and needs exploration is the study of sudden bursts of large positive and negative values the sample paths of non-linear processes generally produce.

There is no unifying theory that is applicable to all non-linear systems and consequently the study of such systems has to be restricted to special classes of non-linear models. Volterra series expansions or polynomial systems are one such class. It is well known that Volterra series expansions are known to be the most general non-linear representation for any stationary sequence.

Volterra series expansions have found significant applications in signal processing such as image, video, and speech processing where the non-linearities are mild enough to allow approximations by low order polynomial approximations. In view of these facts, the study of the extremal properties of finite order Volterra series expansions would be highly valuable in better understanding the extremal properties of non-linear processes as well as understanding the order of identification and adequacy of Volterra series, when used as models in signal processing. In fact, such extremal properties may suggest a way of finding finite order Volterra expansions which is consistent with the non-linearities of the observed process.

In this work, we look at the extremal behavior of Volterra series expansions generated by heavy-tailed innovations, via a point process formulation. We also look at some examples of bilinear processes and compare the known results on extremes of such processes with the corresponding adequate Volterra series approximations. A way of determining the proper order of a finite Volterra expansion and hence a finite order polynomial model for the observed non-linear data is also suggested.