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We study so–called augmented GARCH sequences, which include many sub- models of considerable interest, such as polynomial and exponential GARCH. To model the returns of speculative assets, it is particularly important to understand the behaviour of the squares of the observations. The main aim of this paper is to present a strong approximation for the sum of the squares. This will be achieved by an approximation of the volatility sequence with a sequence of blockwise independent random,variables. Furthermore, we derive a necessary and sufficient condition for the existence of a unique (strictly) stationary solution of the general augmented GARCH equations. Also, necessary and sufficient conditions for the finiteness of moments are provided. AMS 2000 Subject Classification: Primary 60F15, Secondary 62M10.

Strong approximation for the sums of squares of augmented GARCH sequences