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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 m T 12(υ)=∑ X (σj+υ)/T and μT 12(υ)=∑ X (σj +υ)/∑1 where all summations are over indices j such that 0<σj , σj +υ≦T for some σj. The functions M T 12(υ) and μ T 12(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 M T 12(υ) and μ T 12(υ) and discuss some specific situations where it is useful to consider these functions.

A simple expression for the characteristic functional of generalized shot noise is developed. Through expansions in terms of functional derivatives this yields expressions for moment functions of all orders. A central limit theorem also follows. Several examples are discussed.

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Covariation of a time series and a point process

On generalized shot noise

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2 results

Covariation of a time series and a point process

On generalized shot noise