Auto Ball Covariance and Correlation for Fixed-Lag Nonlinear Dependence in Time Series

Classical tools for time series dependence analysis are primarily designed for linear dependence and may fail to detect serial structure when a series is uncorrelated but not independent. To address this problem, we propose the auto ball covariance function and the corresponding auto ball correlation function for measuring lag-specific nonlinear dependence in strictly stationary time series taking values in a separable Banach space. The proposed diagnostic uses metric-ball probabilities to measure fixed-lag distributional dependence without moment requirements, making it suitable for vector-, function-, and norm-induced object-valued time series. Under suitable conditions, we show that the proposed measure is zero if and only if the lagged components are independent. We further develop sample versions of the proposed statistics and establish their large-sample properties, including strong consistency under absolute regularity and a fixed-lag null asymptotic law under a finite-range dependence condition on the lagged-pair process. Simulation studies demonstrate that the proposed method performs well in a variety of settings, especially for nonlinear, heavy-tailed time series. A real-data analysis of annual sunspot numbers further illustrates how the proposed diagnostic can reveal nonlinear residual dependence that is not visible from ordinary autocorrelation diagnostics.