Random observations of marked Cox processes. Time insensitive functionals
Consider a random process \(\tau =\{\tau_0, \tau_1, \tau_2,\dots\}\) on the real axis with the respective sequence of marks \(\chi=\{\chi_0, \chi_1, \chi_2,\dots\}\) (attached to \(\tau\)), which are random vectors \(\chi_n= (X_{n1},\dots,X_{nk})\). The latter represent respective alterations of the status of the system under control, such as quantities of vital files within a particular computer operating system. Now assume that there is a vector \(L=(L_1,\dots, L_k)\) of fixed numerical values (levels) which is predefined and ``kept on file'' by the host system. So that if once, i.e. at one of the random epochs, say \(\tau_j\), the sum \(\mathbf A_j =\sum_{i=0}^j \chi_j\succeq L\) (where for two vectors \(M\succeq L\), if there is at least one entry \(m_s\) of \(M\) which is greater than or equal to \(L_s\); otherwise \(M\prec L\)) gets in \(\succeq\)-relationship with \(L\) for the first time, then the system can be subject to a substantial damage. Some of the components of the vector \(V\) can be taken as \(\infty\). They will correspond to passive components of \(\mathbf A=(\mathbf A_1,\dots, \mathbf A_k)\), provided that at any moment of time any ``surges'' can become arbitrarily large, but finite. The rest of \(\mathbf A\) is referred to as active components. One of the basic models includes a marked delayed renewal process with ``position independent marking''. In this particular case, the increments of \(\mathbf A\): \(\chi_0= \mathbf A_0\), \(\chi_1= \mathbf A_1- \mathbf A_0\), \dots are nonnegative random variables. \textit{J. H. Dshalalow} [J. Appl. Math. Stochastic Anal. 10, No. 4, 355--361 (1997; Zbl 0896.60056)] considered a multivariate marked process with three active and arbitrarily many passive components of \(\mathbf A\). This paper is organized as follows. Section 3 contains the main results on fluctuations for multivariate renewal processes from Dshalalow (loc. cit.). Section 4 introduces generalized Poisson and Cox processes. Section 5 treats delayed renewal processes embedded in generalized Poisson and Cox processes, thereby representing partial observations of such Cox processes. These studies are continued by the authors [J. Math. Anal. Appl. 293, No. 1, 14--27 (2004; Zbl 1047.60005)] leading to time sensitive functionals of these processes and their applications to physics and computer science.
scientific article; zbMATH DE number 2082124
scientific article