Limit theory for some non-linear time series models including GARCH and stochastic volatility models

During the past several years heavy-tailed phenomena have attracted the interest of researchers in time series analysis, extreme value theory, econometrics, telecommunications, and various other fields. The need to consider time series models with heavy-tailed distributions arises from the observation that traditional models of applied probability theory fail to describe jumps, bursts, rapid changes and other erratic behavior of various real-life time series. Heavy-tailed distributions have been considered in the financial time series literature for some time. This includes the GARCH processes whose marginal distributions can have surprisingly heavy (Pareto-like) tails. There is plenty of empirical evidence (see for example Embrechts et al. [7] and the references cited therein) that financial log-return series of stock indices, share prices, exchange rates, etc., can be reasonably modeled by processes with infinite 3rd, 4th or 5th moments. In order to detect non-linearities, the econometrics literature often recommends to consider not only the time series itself but also powers of the absolute values. This can lead to some serious problems: if we accept the premise that the underlying time series has infinite 2nd, 3rd, 4th,... moments, we have to think about the interpretation of the classical tools of time series in this setting. Indeed