Central limit theorem for anomalous scaling due to correlations

Area of Science:

• Probability theory
• Statistical physics
• Stochastic processes

Background:

• Central Limit Theorems (CLTs) typically apply to sums of independent random variables.
• Understanding the behavior of correlated random variables is crucial for modeling complex systems.
• Anomalous scaling and diffusion are observed in various physical phenomena.

Purpose of the Study:

• To derive a central limit theorem for the probability distribution of sums of critically correlated random variables.
• To characterize processes with shared asymptotic anomalous scaling.
• To establish a connection between correlated and uncorrelated random variable sums.

Main Methods:

• Derivation of a novel central limit theorem for correlated random variables.
• Establishing a correspondence with the Lévy-Gnedenko uncorrelated case.
• Mapping correlated anomalous diffusion processes onto Lévy diffusion.

Main Results:

• The derived theorem characterizes diverse processes with identical asymptotic anomalous scaling.
• A direct mapping is established between correlated anomalous diffusion and Lévy diffusion.
• The nonstandard multiplicative structure of the characteristic function determines partial sum correlations from global scaling.

Conclusions:

• The central limit theorem provides a unified framework for understanding sums of correlated random variables.
• The findings offer new insights into anomalous diffusion and scaling phenomena.
• This work bridges the gap between correlated and uncorrelated stochastic processes.