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Covariation of a time series and a point process

Published online by Cambridge University Press:  14 July 2016

J. S. Willie
J. S. Willie*
Affiliation:
Bell Laboratories
*
∗Postal address: 2F-18, Bell Laboratories, 11900 N. Pecos St., Denver, CO 80234, U.S.A.
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Abstract

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.

Keywords

ASYMPTOTIC THEORYCONFIDENCE REGIONSMOMENT FUNCTIONSSTATIONARY PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 19 , Issue 3 , September 1982 , pp. 609 - 618
Copyright
Copyright © Applied Probability Trust 1982 

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References

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