Let {X(t), N(t)}, – ∞< t <∞, be a stationary bivariate stochastic process where X(t) is an ordinary time series and N(t) is an orderly point process counting the number of points in (0, t]. Suppose values of {X(t), N(t)} are available for 0< t ≦T and let σ1, σ2, ···, σ N(T) denote the jump points of N(t) in (0, T]. For |v| < T, define mT12(υ)=∑ X (σj+υ)/T and μT12(υ)=∑ X (σj+υ)/∑1 where all summations are over indices j such that 0<σj, σj+υ≦T for some σj. The functions MT12(υ) and μT12(υ) are often useful in analyzing the covariation of the time series and point process. In this paper, we shall develop some statistical properties of the functions MT12(υ) and μT12(υ) and discuss some specific situations where it is useful to consider these functions.
