2015/09/24 by Alexander Marynych, Marynych, Alexander
Mathematics · #60F05 #60K05 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #advanced mathematical theories #math.PR #msc:60F05 #msc:60K05
paper · pdf · doi:10.48550/arxiv.1509.07321
arxiv created 2015/09/24 · openalex publication_date 2015/09/24 · arxiv updated 2015/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ≥ 0Xk+1(t-ξ1-…-ξk)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 Iksanov, Marynych and Meiners (2015+).