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 $\alpha=2$.

Overview of The Gaussian Central Limit Theorem for a Stationary Time Series with Infinite Variance

The paper, "The Gaussian Central Limit Theorem for a Stationary Time Series with Infinite Variance" by Muneya Matsui and Thomas Mikosch, investigates the application of the Gaussian Central Limit Theorem (CLT) in the context of stationary time series where the marginal distributions exhibit infinite variance. This paper addresses a complex facet of probability theory, examining the conditions required to achieve a Gaussian limit in such settings.

Key Findings and Methodology

In scenarios where the individual elements of a time series are identically distributed, independent, and regularly varying with a tail index of $\alpha = 2$, previous research has shown that the Gaussian CLT results hold. Matsui and Mikosch extend these findings to dependent time series by assuming a stricter condition of sequential regular variation. This assumption includes the segmentation of time series data into asymptotically independent blocks.

The authors use classical results regarding row-wise independent and identically distributed (iid) triangular arrays to support their analysis. They establish necessary and sufficient conditions for the independence of block sums by employing large deviation techniques. These are demonstrated across various time series models, such as m-dependent sequences, linear processes, stochastic volatility processes, and solutions to affine stochastic recurrence equations—all possessing regularly varying marginal distributions with infinite variance.

Implications of the Research

The theoretical implications of Matsui and Mikosch’s work are significant, primarily in understanding the behavior of complex stochastic processes exhibiting heavy tails and infinite variance. By establishing a connection between regular variation, large deviation principles, and the Gaussian CLT, the authors provide a comprehensive framework for assessing the convergence of sums of dependent random variables to a Gaussian distribution.

Practically, this paper extends the toolkit available to statisticians and researchers working with financial time series, telecommunications data, and environmental data, where such heavy-tailed processes are frequently encountered. In applied fields, being able to invoke the Gaussian CLT under broader conditions can simplify the usage of asymptotic results for inference and hypothesis testing.

Anticipated Future Directions

The scope of future research includes exploring different structures within dependent time series that satisfy the conditions outlined by Matsui and Mikosch and investigating other forms of dependencies and their impact on central limit theorem applicability. Furthermore, examining the robustness of their findings under varying assumptions about mixing conditions and extending the analysis to multi-dimensional time series would provide additional depth.

Additionally, this theoretical framework could guide new simulation studies aimed at verifying the practical limits of these mathematical results under realistic conditions. Enhancing our understanding of how these theoretical constructs apply to complex real-world data remains a pressing objective for future studies.

In conclusion, this paper contributes to the broader understanding of Gaussian approximations in non-standard conditions, such as infinite variance, offering valuable insights into the behavior of stationary time series. The interdisciplinary potential of these results underscores the paper's importance within statistical theory and applied domains alike.