This article introduces new methods for the analysis of cyclostationary time series with infinite variance. Traditional cyclostationary analysis, based on periodically correlated (PC) processes, relies on the autocovariance function (ACVF). However, the ACVF is not suitable for data exhibiting a heavy-tailed distribution, particularly with infinite variance. Thus, we propose a novel framework for the analysis of cyclostationary time series with heavy-tailed distribution, utilizing the fractional lower-order covariance (FLOC) as an alternative to covariance. This leads to the introduction of two new autodependence measures: the periodic fractional lower-order autocorrelation function (peFLOACF) and the periodic fractional lower-order partial autocorrelation function (peFLOPACF). These measures generalize the classical periodic autocorrelation function (peACF) and periodic partial autocorrelation function (pePACF), offering robust tools for analyzing infinite-variance processes. Two practical applications of the proposed measures are explored: a portmanteau test for testing dependence in cyclostationary series and a method for order identification in periodic autoregressive (PAR) and periodic moving average (PMA) models with infinite variance. Both applications demonstrate the potential of new tools, with simulations validating their efficiency. The methodology is further illustrated through the analysis of real-world air pollution data, which showcases its practical utility. The results indicate that the proposed measures based on FLOC provide reliable and efficient techniques for analyzing cyclostationary processes with heavy-tailed distributions.

翻译：本文提出了分析具有无穷方差的重尾分布循环平稳时间序列的新方法。传统基于周期相关过程的循环平稳分析依赖于自协方差函数，但不适用于重尾分布数据，尤其是无穷方差情形。因此，我们构建了重尾分布循环平稳时间序列分析的新框架，采用分数低阶协方差替代协方差函数，由此引入两种新的自依赖度量：周期分数低阶自相关函数与周期分数低阶偏自相关函数。这些度量推广了经典周期自相关函数与周期偏自相关函数，为无穷方差过程分析提供了稳健工具。我们探索了所提度量的两种实际应用：循环平稳序列依赖性检验的端口曼特检验，以及具有无穷方差的周期自回归与周期滑动平均模型的阶数识别方法。两种应用均展现了新工具的潜力，仿真实验验证了其有效性。通过真实空气污染数据的分析进一步展示了方法的实用价值。结果表明，基于分数低阶协方差的度量方法为重尾分布循环平稳过程分析提供了可靠高效的技术手段。