The Gaussian Central Limit Theorem for a Stationary Time Series With Infinite Variance
研究了严格平稳时间序列在无限方差但极限分布为高斯分布时的中心极限定理，通过块分割和独立块和条件，利用大偏差结果推导了必要充分条件，适用于多种依赖序列。
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 i.i.d. case, a well‐known sufficient condition for this central limit theorem is regular variation of the marginal distribution with tail index . In the dependent case, we assume the stronger condition of sequential regular variation of the time series with tail index . We assume that a sample of size from this time series can be split into blocks of size such that as and that the block sums are asymptotically independent. Then we apply classical central limit theory for row‐wise i.i.d. 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 ‐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 .