Statistics - Levy Laws and 1/f noise

A unified and universal explanation for Lévy laws and 1/f noises

Lévy laws and 1/f noises are shown to emerge uniquely and universally from a general model of systems which superimpose the transmissions of many independent stochastic signals. The signals are considered to follow, statistically, a common—yet arbitrary—generic signal pattern which may be either stationary or dissipative. Each signal is considered to have its own random transmission amplitude and frequency. We characterize the amplitude-frequency randomizations which render the system output's stationary law and power-spectrum universal—i.e., independent of the underlying generic signal pattern. The classes of universal stationary laws and power spectra are shown to coincide, respectively, with the classes of Lévy laws and 1/f noises—thus providing a unified and universal explanation for the ubiquity of these classes of “anomalous statistics” in various fields of science and engineering.