Expected correlation in time-series analysis

Time-series analysis often involves the characterization of order or predictability, qualities that are related to internal structure and autocorrelation. Investigating a recently proposed algorithm for solving a density prediction task, we demonstrate that if the same system can be viewed on multiple time scales, there is an inevitable degree of expected order and predictability that increases as the system size grows. In particular, we introduce bounds on the expected second-order structure function and auto-correlation function of a time series where multiple observation scales are available, and conclude with a lower bound on the expected correlation time. Such a lower bound shows that a certain degree of autocorrelation is inevitable, and therefore provides a stronger baseline against which to measure anomalously large or small order and predictability.