Functional Limit Theorem for the Sums of PSI-Processes with Random Intensities

Abstract

We consider a sequence of i.i.d. random variables, (ξ)=(ξi)i=0,1,2,⋯, Eξ0=0, Eξ02=1, and subordinate it by a doubly stochastic Poisson process Π(λt), where λ≥0 is a random variable and Π is a standard Poisson process. The subordinated continuous time process ψ(t)=ξΠ(λt) is known as the PSI-process. Elements of the triplet (Π,λ,(ξ)) are supposed to be independent. For sums of n, independent copies of such processes, normalized by n, we establish a functional limit theorem in the Skorokhod space D[0,T], for any T>0, under the assumption E|ξ0|2h<∞ for some h>1/γ2. Here, γ∈(0,1] reflects the tail behavior of the distribution of λ, in particular, γ≡1 when Eλ<∞. The limit process is a stationary Gaussian process with the covariance function Ee−λu, u≥0. As a sample application, we construct a martingale from the PSI-process and establish a convergence of normalized cumulative sums of such i.i.d. martingales.