  The aim of this paper is to provide conditions which ensure that the affinely
transformed partial sums of a strictly stationary process converge in
distribution to an infinite variance stable distribution. Conditions for this
convergence to hold are known in the literature. However, most of these results
are qualitative in the sense that the parameters of the limit distribution are
expressed in terms of some limiting point process. In this paper we will be
able to determine the parameters of the limiting stable distribution in terms
of some tail characteristics of the underlying stationary sequence. We will
apply our results to some standard time series models, including the GARCH(1,
1) process and its squares, the stochastic volatility models and solutions to
stochastic recurrence equations.
