Power-law random walks

TL;DR

This paper investigates the properties of power-law random walks with steps following a q-Gaussian distribution, providing explicit representations and geometric interpretations depending on the parameter q.

Contribution

It introduces new explicit stochastic representations for the walk when q>1 and a geometric interpretation as a projection of isotropic walks when q<1.

Findings

Explicit stochastic representation for q>1 case.

Geometric interpretation as projection for q<1.

Enhanced understanding of distributional properties of power-law random walks.

Abstract

We present some new results about the distribution of a random walk whose independent steps follow a $q-$Gaussian distribution with exponent $\frac{1}{1-q}; q \in \mathbb{R}$. In the case $q>1$ we show that a stochastic representation of the point reached after $n$ steps of the walk can be expressed explicitly for all $n$. In the case $q<1,$ we show that the random walk can be interpreted as a projection of an isotropic random walk, i.e. a random walk with fixed length steps and uniformly distributed directions.