Angular processes related to Cauchy random walks

TL;DR

This paper investigates the angular processes of Cauchy random walks in Euclidean and non-Euclidean spaces, revealing new transformations, explicit distributions, and connections to Fibonacci numbers and the golden ratio.

Contribution

It introduces novel non-linear transformations of Cauchy variables that preserve their distribution and analyzes continued fractions, linking them to Fibonacci numbers and convergence to the golden ratio.

Findings

Derived explicit distributions for transformed Cauchy variables.

Proved that n-stage continued fractions are Cauchy distributed with Fibonacci-related parameters.

Showed convergence of the process to the golden ratio.

Abstract

We study the angular process related to random walks in the Euclidean and in the non-Euclidean space where steps are Cauchy distributed. This leads to different types of non-linear transformations of Cauchy random variables which preserve the Cauchy density. We give the explicit form of these distributions for all combinations of the scale and the location parameters. Continued fractions involving Cauchy random variables are analyzed. It is shown that the $n$-stage random variables are still Cauchy distributed with parameters related to Fibonacci numbers. This permits us to show the convergence in distribution of the sequence to the golden ratio.