On multivariate delayed recurrent processes. (English) Zbl 1078.60036

This paper introduces and analyzes a class of delayed renewal point processes \(\tau=\{\tau_0, \tau_1, \tau_2,\dots \}\) marked by a multivariate recurrent process \((\tau, S)=\sum_{n=0}^{\infty} \chi_n \varepsilon_{\tau_n}\) (where \(\varepsilon\) is a point mass) with position dependent marking. Let \(S=\{S_0, S_1, S_2,\dots \}\) be an \((k+m)\)-variate with the associated increments \(\chi_0= S_0\), \(\chi_1 = S_1 - S_0\), \(\chi_2 = S_2 - S_1\),…, being independent random vectors, of which \(\chi_1\), \(\chi_2\), …be identically distributed. The first \(k\) of the \(k+m\) components of \(S\) called active are watched close, especially at the crossing. The rest of the components are referred to passive and they are just to be registered. The fixed levels to be possibly crossed are denoted by \(L=(L_1,\dots ,L_k)\).
In connection with crossing, the authors introduce the first passage index \(\rho=\inf\{n: S_n\succ L\}\) (where \(S_n\succ L\), if there is at least one entry \(S_n^i\) of the first \(k\) of \(S_n\), which is greater than \(L_s\); otherwise, \(S_n\preceq L\)) and call \(\tau_{\rho}\) the passage time, while \(S_{\rho}\) is called the first excess value of \((\tau, S)\). One of the objectives of these studies is to find the joint probability distribution of \(\rho\), \(\tau_{\rho}\) and \(S_{\rho}\). Furthermore, the so-called exit time \(\tau_{\rho-1}\) and the corresponding value of the recurrent process at the exit time, \(S_{\rho-1}\), are of great interest. A joint probability distribution of \(\rho\), \(\tau_{\rho}\), \(\tau_{\rho-1}\), \(S_{\rho-1}\) and \(S_{\rho}\) is obtained in a closed form. The results for the marked recurrent process are applied to the generalized marked Poisson process \(\Pi\) (with real-valued marks), observed at random times \(\tau\) forming a delayed renewal process. The results obtained are illustrated by various examples and special cases.


60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60K05 Renewal theory
60G57 Random measures