Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes

TL;DR

This paper derives a closed-form probability distribution function for the Euclidean distance between two independent telegraph processes with finite velocities, starting at the same point and governed by Poisson processes, providing new analytical insights.

Contribution

It provides the first explicit closed-form expression for the distribution of the distance between two independent telegraph processes at any time.

Findings

Closed-form distribution function derived

Numerical results illustrating the distribution

Analytical insights into telegraph process distances

Abstract

Consider two independent Goldstein-Kac telegraph processes $X_1(t)$ and $X_2(t)$ on the real line $\Bbb R$. The processes $X_k(t), \; k=1,2,$ are performed by stochastic motions at finite constant velocities $c_1>0, \; c_2>0,$ that start at the initial time instant $t=0$ from the origin of the real line $\Bbb R$ and are controlled by two independent homogeneous Poisson processes of rates $\lambda_1>0, \; \lambda_2>0$, respectively. Closed-form expression for the probability distribution function of the Euclidean distance $$\rho(t) = |X_1(t) - X_2(t) |, \qquad t>0,$$ between these processes at arbitrary time instant $t>0$, is obtained. Some numerical results are presented.