Asymptotics of random processes with immigration I: Scaling limits

摘要

Let (X-1, xi(1)), (X-2, xi(2)),... be i.i.d. copies of a pair (X, xi) where X is a random process with paths in the Skorokhod space D[0, infinity) and xi is a positive random variable. Define S-k := xi(1) + ... + xi(k), k is an element of N-0 and Y(t) := Sigma(k >= 0) Xk+1(t - S-k)1({sk <= t}), t >= 0. We call the process (Y (t))(t >= 0) random process with immigration at the epochs of a renewal process. We investigate weak convergence of the finite -dimensional distributions of (Y (ut))(u>0) as t -> infinity. Under the assumptions that the covariance function of X is regularly varying in (0, infinity) x (0, infinity) in a uniform way, the class of limiting processes is rather rich and includes Gaussian processes with explicitly given covariance functions, fractionally integrated stable Levy motions and their sums when the law of xi belongs to the domain of attraction of a stable law with finite mean, and conditionally Gaussian processes with explicitly given (conditional) covariance functions, fractionally integrated inverse stable subordinators and their sums when the law of xi belongs to the domain of attraction of a stable law with infinite mean.