A two-parameter random walk with approximate exponential probability distribution

TL;DR

This paper analyzes a non-Markovian one-dimensional random walk with two parameters, deriving a closed-form joint probability distribution and showing it approximately follows an exponential family distribution.

Contribution

It provides a closed-form expression for the joint distribution of position and reversals in a two-parameter non-Markovian random walk, revealing its exponential family structure.

Findings

Derived a closed-form joint probability distribution p_n(x,k).

Showed the distribution approximately belongs to the exponential family.

Enhanced understanding of non-Markovian random walk behavior.

Abstract

We study a non-Markovian random walk in dimension 1. It depends on two parameters eps_r and eps_l, the probabilities to go straight on when walking to the right, respectively to the left. The position x of the walk after n steps and the number of reversals of direction k are used to estimate eps_r and eps_l. We calculate the joint probability distribution p_n(x,k) in closed form and show that, approximately, it belongs to the exponential family.