1∕ f ^β noise for scale-invariant processes: how long you wait matters

We study the power spectrum which is estimated from a nonstationary signal. In particular we examine the case when the signal is observed in a measurement time window [t_w, t_w + t_m], namely the observation started after a waiting time t_w, and t_m is the measurement duration. We introduce a generalized aging Wiener–Khinchin theorem which relates between the spectrum and the time- and ensemble-averaged correlation functions for arbitrary t_m and t_w. Furthermore we provide a general relation between the non-analytical behavior of the scale-invariant correlation function and the aging 1∕f^β noise. We illustrate our general results with two-state renewal models with sojourn times’ distributions having a broad tail.