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Universal generation of statistical self-similarity: a randomized central limit theorem
1Department of Technology Management, Holon Institute of Technology, Holon 58102, Israel. eliazar@post.tau.ac.il
Physical Review Letters
|August 8, 2009
Summary
A universal mechanism generates fractal patterns in random processes by superimposing signals with common patterns but varied parameters. This leads to anomalous diffusion and 1/f noise, akin to a "randomized central limit theorem" for fractal statistics.
Area of Science:
- Complex Systems Science
- Statistical Physics
- Stochastic Processes
Background:
- Fractality and statistical self-similarity are observed in numerous natural and artificial systems.
- Understanding the underlying mechanisms generating these fractal properties in random processes is a key challenge.
Purpose of the Study:
- To establish a universal mechanism for generating statistical self-similarity (fractality) in random processes.
- To characterize the conditions under which superimposed stochastic signals lead to fractal outputs.
Main Methods:
- Consideration of a generic system that superimposes independent stochastic signals with shared patterns but randomized transmission parameters (amplitude, frequency, initiation epoch).
- Mathematical characterization of parameter randomizations that yield statistically self-similar outputs universally.
Main Results:
- A universal mechanism for generating statistical self-similarity in random processes is established.
- Statistically self-similar outputs with finite variance exhibit anomalous diffusion (power-law temporal variance growth) and 1/f noise (power-law power spectra).
Conclusions:
- The presented mechanism acts as a
- randomized central limit theorem
- for fractal statistics of random processes.
- This framework provides a unified explanation for fractality, anomalous diffusion, and 1/f noise in systems composed of superimposed stochastic signals.
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