### 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 X

_{1}, X

_{2}, ... be random elements of the Skorokhod space D(R) and ξ

_{1}, ξ

_{2}, ... positive random variables such that the pairs (X

_{1}, ξ

_{1}), (X

_{2},ξ

_{2}), ... are independent and identically distributed. The random process

_{k≥0}X

_{k+1}(t-ξ

_{1}-...-ξ

_{k}) 1

_{{ξ1+...+ξk≤t}}, t∈R,

_{1}is nonlattice and has finite mean while the process X

_{1}decays sufficiently fast, we prove weak convergence of (Y(u+t))

_{{u∈R}}as t→∞ on D(R) endowed with the J

_{1}-topology. The present paper continues the line of research initiated in [2,3]. Unlike the corresponding result in [3] arbitrary dependence between X

_{1}and ξ

_{1}is allowed.