Modelling of random processes and fields based on randomized piecewise constant replenishment of discrete argument processes and Palm flows with summation of their trajectories

Abstract The paper presents a generalization of the well-known method of modelling a continuous argument process based on randomized piecewise constant replenishment of the discrete argument process. It is proposed to additionally summarize the trajectories of a piecewise constant process in the case the distributions of the initial process are infinitely divisible. The behavior of continuous argument processes is studied when the number of terms tends to infinity. The conditions under which the sequence of processes constructed this way weakly converges in the Skorokhod and L p spaces are presented and the properties of limiting processes are studied. In addition, an algorithm for modelling homogeneous random fields with piecewise constant realizations based on the Palm flows and summation of realizations is proposed.