Time sensitive functionals of marked Cox processes
Imagine a complex system with various components to fail, such as an Internet server, an operating system, or complex software with multiple subroutines. Some of the most vital parts of the system need to be observed from time to time. Notice that often phenomena cannot be observed continuously and information is usually collected at some random epochs of time \(\tau=\{\tau_1, \tau_2,\dots\}\). The frequency of such data can be increased at some expense but in some cases we have no control. There are classes of continuous time parameter multivariate processes where the collection of data takes place at the moments of a renewal process and the marks can retain some of the renewal properties (with no independence of their increments of their positions). Examples of such processes are multivariate stationary Poisson and Cox processes [see [1] the authors, J. Math. Anal. Appl. 293, 1--13 (2004; Zbl 1052.60036)]. This paper continues studies on a class of multivariate marked Cox processes \(C_t\) observed by a delayed renewal process initiated in [1]. The goal of the paper is to connect the continuous time parameter process \(C_t\) with \(C_{\tau}\) for which closed-form transforms were presented in [1]. The authors establish a bridge between time sensitive and time insensitive functionals initiated in [1]. The time sensitive functional does not fully restore the information data lost due to restricted observations. The paper is organized as follows. Section 2 provides some auxiliary results (Lemmas 1 and 2). In Section 3 the main results concerning time sensitive functionals are proved. In particular, analytically tractable formulas for functionals of \(C_{\tau}\) are derived.
scientific article; zbMATH DE number 2082125