Asymptotics of random processes with immigration II: Convergence to stationarity

作者：Iksanov Alexander^*; Marynych Alexander^*; Meiners Matthias

来源：Bernoulli, 2017, 23(2): 1279-1298.

DOI：10.3150/15-BEJ777

• random point process
• renewal shot noise process
• stationary renewal process
• weak convergence in the Skorokhod space

Let X-1, X-2,... be random elements of the Skorokhod space D(R) and xi(1), xi(2),... positive random variables such that the pairs (X-1, xi(1).), (X-2, xi(2)),... are independent and identically distributed. We call the random process (Y(t))(t is an element of R) defined by Y(t) := Sigma(k >= 0) Xk+1(t - xi(1) - ... -xi(k))1({xi 1+...+xi k <= t}), t is an element of R random process with immigration at the epochs of a renewal process. Assuming that X-k and xi(k) are independent and that the distribution xi(1) is nonlattice and has finite mean we investigate weak convergence of (Y(t))(t is an element of R) as t -> infinity in D(R) endowed with the J(1)-topology. The limits are stationary processes with immigration.

• 出版日期2017-5