Marked Point Process

A marked point process with mark space  is a double sequence

of -valued random variables and -valued random variables  defined on a probability space  such that  is a simple point process (SPP) and:

1.  for ;

2.  for .

Here,  denotes probability,  denotes the so-called irrelevant mark which is used to describe the mark of an event that never occurs, and .

This definition is similar to the definition of an SPP in that it describes a sequence of time points marking the occurrence of events. The difference is that these events may be of different types where the type (i.e., the mark) of the th event is denoted by . Note that, because of the inclusion of the irrelevant mark , marking will assign values  for all --even when , i.e., when the th event never occurs (Jacobsen 2006).

REFERENCES:

Jacobsen, M. Point Process Theory and Applications: Marked Point and Piecewise Deterministic Process. Boston: Birkhäuser, 2006.