Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes. (December 2014)

• Record Type: Journal Article
• Authors: Kolesnik, Alexander D.
• Is Part Of: Advances in applied probability. Volume 46:Number 4(2014)
• Page Start: 1172
• Page End: 1193
• Publication Date: 2014-12
• Subjects: Telegraph process, -- telegraph equation, -- persistent random walk, -- probability distribution function, -- Euclidean distance; 60K35, -- 60J60, -- 60J65, -- 82C41, -- 82C70; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1239/aap/1418396248
• Languages: English
• ISSNs: 0001-8678

Abstract

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.