Asymptotics of random processes with immigration II: convergence to stationarity

2013/11/27 by Alexander Iksanov, Iksanov, Alexander, Alexander Marynych +3
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1311.6923

20 pages, accepted for publication in Bernoulli

arxiv created 2015/10/09 · arxiv updated 2015/10/12

Abstract

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. We call the random process (Y(t))_{t ∈ ℝ} defined by Y(t):=∑_{k ≥ 0}X_k+1(t-ξ₁-…-ξ_k)1_\ξ₁+…+ξ_k≤ t\, t∈ℝ random process with immigration at the epochs of a renewal process. Assuming that X_k and ξ_k are independent and that the distribution of ξ₁ is nonlattice and has finite mean we investigate weak convergence of (Y(t))_t∈ℝ as t→∞ in D(ℝ) endowed with the J₁-topology. The limits are stationary processes with immigration.

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