Counting Process

A counting process is a stochastic process whose sample paths are nondecreasing, right-continuous, and nonnegative integer-valued, with . It models a stream of events by taking to be the number that have occurred by time . Thus, for , the increment

counts the events occurring in the interval .

If every jump has size one, the jump times of a counting process define a simple point process. Conversely, a locally finite simple temporal point process determines a counting process by counting its points up to each time (Daley and Vere-Jones 2003).