System of telegraph particles with finite moments of the first collision instant of particles

Abstract

This paper deals with a system of interacting telegraph particles starting with different positions on a straight line. It is well-known that the instant of the first collision of two telegraph particle, that starts from different points on a line, has an infinite expectation. Our goal is to find a sufficient number of particles of the system such that the minimum of the first collision instants for these particles has finite $n$th order moments. In particular, finite expectation, finite variance, etc. However, the distribution of this minimum depends on first collisions of all pairs of adjacent particles, and these collisions are dependent random variables, which introduces some difficulties in the analysis.