The Gaussian central limit theorem for a stationary time series with infinite variance
Muneya Matsui, Thomas Mikosch
TL;DR
The paper investigates Gaussian central limit theorems for strictly stationary time series with infinite variance in the boundary case $\\alpha=2$, using sequential regular variation and a block-splitting approach to reduce the problem to a triangular array of iid block sums. By enforcing a mixing condition and applying precise large-deviation arguments tied to the spectral tail process $\\Theta_t$, the authors derive the CLT for a broad class of dependent models, and provide explicit formulas for the asymptotic variance in terms of $\\Theta_t$ for $m$-dependent sequences, linear processes, stochastic volatility models, and affine stochastic recurrence equations (Kesten-Goldie and Grincevičius-Grey). The results extend Gaussian limit theory to infinite-variance time series under dependence, highlighting the role of extremes and large-deviation structure in achieving normal limits. This advances understanding of fluctuations in heavy-tailed time series and offers practical tools for modeling in finance and econometrics where infinite-variance dynamics are present.
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 $α=2$. In the dependent case we assume the stronger condition of sequential regular variation of the time series with tail index $α=2$. 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 $α=2$.