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.