1/f Noise and the Infrared Catastrophe

It is generally assumed that stochastic processes exhibiting 1/f noise are affected with the so-called infrared catastrophe. We present an intermittent stochastic process generating 1/f noise which avoids this problem.

💡 Research Summary

The paper addresses a long‑standing paradox in the theory of 1/f (flicker) noise: while many physical, biological, and technological systems exhibit a power spectral density (PSD) that scales as S(f) ∝ 1/f over several decades, the mathematical integration of such a spectrum down to zero frequency diverges, leading to the so‑called “infrared catastrophe.” Traditional explanations avoid this divergence by imposing an arbitrary low‑frequency cutoff, by assuming a finite observation time, or by superposing a broad distribution of Lorentzian spectra with an ad‑hoc lower bound on relaxation times. These approaches, however, lack a clear physical mechanism and often fail to reproduce the statistical properties observed in real data, such as heavy‑tailed waiting‑time distributions.

To resolve this issue, the authors propose an intermittent stochastic process that naturally generates 1/f‑type spectra while guaranteeing a finite total power. The process consists of two elementary ingredients:

1. Renewal events with power‑law inter‑event times. The waiting time τ between successive events follows a heavy‑tailed probability density
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