This paper investigates the asymptotic distribution of a wavelet-based NKK periodogram constructed from least absolute deviations (LAD) harmonic regression at a fixed resolution level. Using a wavelet representation of the underlying time series, we analyze the probabilistic structure of the resulting periodogram under long-range dependence. It is shown that, under suitable regularity conditions, the NKK periodogram converges in distribution to a nonstandard limit characterized as a quadratic form in a Gaussian random vector, whose covariance structure depends on the memory properties of the process and on the chosen wavelet filters. This result establishes a rigorous theoretical foundation for the use of robust wavelet-based periodograms in the spectral analysis of long-memory time series with heavy-tailed inovations.

翻译：本文研究了在固定分辨率水平下，由最小绝对偏差（LAD）谐波回归构造的小波基NKK周期图的渐近分布。利用基础时间序列的小波表示，我们分析了在长程相依性下所得周期图的概率结构。结果表明，在适当的正则性条件下，NKK周期图依分布收敛于一个非标准极限，该极限被表征为一个高斯随机向量的二次型，其协方差结构取决于过程的记忆特性以及所选的小波滤波器。这一结果为在具有重尾新息的长记忆时间序列谱分析中使用基于稳健小波的周期图建立了严格的理论基础。