Weak convergence of renewal shot noise processes in the case of slowly varying normalization

We investigate weak convergence of finite-dimensional distributions of a renewal shot noise process (Y(t))t≥0(Y(t))t≥0 with deterministic response function hh and the shots occurring at the times 0=S02r>2 we use a strong approximation argument to show that the random fluctuations of Y(s)Y(s) occur on the scale s=t+g(t,u)s=t+g(t,u) for u∈[0,1]u∈[0,1], as t→∞t→∞, and, on the level of finite-dimensional distributions, are well approximated by the sum of a Brownian motion and a Gaussian process with independent values (the two processes being independent). The scaling function gg above depends on the slowly varying factor of hh. If, for instance, limt→∞t1/2h(t)∈(0,∞)limt→∞t1/2h(t)∈(0,∞), then g(t,u)=tug(t,u)=tu.