Multifractality of the multiplicative autoregressive point processes

Multiplicative processes and multifractals have earned increased popularity in applications ranging from hydrodynamic turbulence to computer network traffic, from image processing to economics. We analyse the multifractality of the recently proposed point process models generating the signals exhibiting 1/f^b noise. The models may be used for modeling and analysis of stochastic processes in different systems. We show that the multiplicative point process models generate multifractal signals, in contrast to the formally constructed signals with 1/f^b noise and signals consisting of sum of the uncorrelated components with a wide-range distribution of the relaxation times.

💡 Research Summary

The paper addresses a fundamental question in the study of stochastic processes that exhibit 1/f^β noise: can a simple generative model produce not only the correct power‑law spectrum but also the rich multifractal scaling observed in many natural and engineered systems? The authors begin by reviewing two conventional approaches. The first constructs a signal directly in the frequency domain, imposing a prescribed S(f)∝f^{-β}. While this guarantees the desired spectral exponent, the resulting time series is essentially linear and lacks any hierarchical variability. The second approach sums a large number of independent exponential relaxations with a broad distribution of time constants. This method can also reproduce a 1/f^β spectrum, but because the components are statistically independent, the aggregate signal does not display non‑trivial multifractal behavior.

To overcome these limitations, the authors propose a multiplicative autoregressive point‑process model. Events occur at random times, and the inter‑event interval τ_n evolves according to a stochastic recursion: \