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

We consider a sequence of i.i.d. random variables, ( ξ ) = ( ξ i ) i = 0 , 1 , 2 , ⋯ , E ξ 0 = 0 , E ξ 0 2 = 1 , and subordinate it by a doubly stochastic Poisson process Π ( λ t ) , where λ ≥ 0 is a random variable and Π is a standard Poisson process. The subordinated continuous time process ψ ( t ) = ξ Π ( λ t ) 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 | ξ 0 | 2 h < ∞ for some h > 1 / γ 2 . Here, γ ∈ ( 0 , 1 ] reflects the tail behavior of the distribution of λ , in particular, γ ≡ 1 when E λ < ∞ . The limit process is a stationary Gaussian process with the covariance function E e − λ u , 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.