The Gaussian central limit theorem for a stationary time series with infinite variance

Abstract

We consider a borderline case: the central limit theorem for a strictly stationary time series with infinite variance but a Gaussian limit. In the iid case a well-known sufficient condition for this central limit theorem is regular variation of the marginal distribution with tail index $\alpha=2$. In the dependent case we assume the stronger condition of sequential regular variation of the time series with tail index $\alpha=2$. We assume that a sample of size $n$ from this time series can be split into $k_n$ blocks of size $r_n\to\infty$ such that $r_n/n\to 0$ as $n\to\infty$ and that the block sums are asymptotically independent. Then we apply classical central limit theory for row-wise iid triangular arrays. The necessary and sufficient conditions for such independent block sums will be verified by using large deviation results for the time series. We derive the central limit theorem for $m$-dependent sequences, linear processes, stochastic volatility processes and solutions to affine stochastic recurrence equations whose marginal distributions have infinite variance and are regularly varying with tail index $\alpha=2$.