  Long Memory Stochastic volatility (LMSV) models capture two standardized
features of financial data: the log-returns are uncorrelated, but their
squares, or absolute values are (highly) dependent and they may have heavy
tails. EGARCH and related models were introduced to model leverage, i.e.
negative dependence between previous returns and future volatility. Limit
theorems for partial sums, sample variance and sample covariances are basic
tools to investigate the presence of long memory and heavy tails and their
consequences. In this paper we extend the existing literature on the asymptotic
behaviour of the partial sums and the sample covariances of long memory
stochastic volatility models in the case of infinite variance. We also consider
models with leverage, for which our results are entirely new in the infinite
variance case. Depending on the nterplay between the tail behaviour and the
intensity of dependence, wo types of convergence rates and limiting
distributions can arise. In articular, we show that the asymptotic behaviour of
partial sums is the same for both LMSV and models with leverage, whereas there
is a crucial difference when sample covariances are considered.
