Heavy-Tail Distribution from Correlation of Discrete Stochastic Process

TL;DR

This paper introduces a stochastic process with memory effects that produces heavy-tailed distributions, demonstrating that memory can be an alternative origin of heavy tails in stochastic systems.

Contribution

It analytically derives a new class of distributions from a discrete process with auto-correlation and explores how memory effects lead to heavy-tailed behaviors.

Findings

Derived a closed-form moment generating function.

Showed convergence of cumulants.

Demonstrated power-law decay from regime switching.

Abstract

We propose a stochastic process driven by the memory effect with novel distributions which include both exponential and leptokurtic heavy-tailed distributions. A class of the distributions is analytically derived from the continuum limit of the discrete binary process with the renormalized auto-correlation. The moment generating function with a closed form is obtained, thus the cumulants are calculated and shown to be convergent. The other class of the distributions is numerically investigated. The combination of the two stochastic processes of memory with different signs under regime switching mechanism does result in behaviors of power-law decay. Therefore we claim that memory is the alternative origin of heavy-tail.