Aging Wiener-Khinchin theorem and critical exponents of 1/f (beta) noise

The power spectrum of a stationary process may be calculated in terms of the autocorrelation function using the Wiener-Khinchin theorem. We here generalize the Wiener-Khinchin theorem for nonstationary processes and introduce a time-dependent power spectrum < S-tm (omega)> where t(m) is the measurement time. For processes with an aging autocorrelation function of the form < I (t) I (t + tau)> = t(gamma) phi(EA) (tau/t), where phi(EA) (x) is a nonanalytic function when x is small, we find aging 1/f (beta) noise. Aging 1/f (beta) noise is characterized by five critical exponents. We derive the relations between the scaled autocorrelation function and these exponents. We show that our definition of the time-dependent spectrum retains its interpretation as a density of Fourier modes and discuss the relation to the apparent infrared divergence of 1/f (beta) noise. We illustrate our results for blinking-quantum-dot models, single-file diffusion, and Brownian motion in a logarithmic potential.