Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes. (December 2014) - Kolesnik, Alexander D.
Abstract : Consider two independent Goldstein-Kac telegraph processes X 1 ( t ) and X 2 ( t ) on the real line ℝ. The processes X k ( t ), k = 1, 2, describe stochastic motions at finite constant velocities c 1 > 0 and c 2 > 0 that start at the initial time instant t = 0 from the origin of ℝ and are controlled by two independent homogeneous Poisson processes of rates λ1 > 0 and λ2 > 0, respectively. We obtain a closed-form expression for the probability distribution function of the Euclidean distance ρ( t ) = | X 1 ( t ) - X 2 ( t )|, t > 0, between these processes at an arbitrary time instant t > 0. Some numerical results are also presented.