A note on convergence to stationarity of random processes with immigration

A. V. Marynych
Theory of Stochastic Processes
Vol.20 (36), no.1, 2015, pp.84-100
Let X1, X2, ... be random elements of the Skorokhod space D(R) and ξ1, ξ2, ... positive random variables such that the pairs (X1, ξ1), (X22), ... are independent and identically distributed. The random process Y(t):=∑k≥0 Xk+1 (t-ξ1-...-ξk) 11+...+ξk≤t}, t∈R, is called random process with immigration at the epochs of a renewal process. Assuming that the distribution of ξ1 is nonlattice and has finite mean while the process X1 decays sufficiently fast, we prove weak convergence of (Y(u+t)){u∈R} as t→∞ on D(R) endowed with the J1-topology. The present paper continues the line of research initiated in [2,3]. Unlike the corresponding result in [3] arbitrary dependence between X1 and ξ1 is allowed.