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), (X₂,ξ₂), ... are independent and identically distributed. The random process Y(t):=∑_k≥0 X_k+1 (t-ξ₁-...-ξ_k) 1_{{ξ1+...+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.