counterprocess

A counterprocess, in probability theory and stochastic processes, refers to a counting process that records the total number of events that have occurred up to any given time. It is a nonnegative, integer-valued, non-decreasing process that starts at zero and increases by one at event times, remaining constant between events. Such processes are typically denoted by N(t) for t ≥ 0.

Key properties and variations: A counting process is often assumed to be adapted to a filtration, and it is right-continuous with left limits. For any two times s ≤ t, the increment N(t) − N(s) counts the number of events in the interval (s, t], and is a nonnegative integer. In the canonical Poisson process, increments over disjoint intervals are independent and follow a Poisson distribution with parameter proportional to the interval length, giving a homogeneous counting process. More general counting processes can have time-varying rates or dependent increments, such as non-homogeneous Poisson processes, renewal processes, or more complex point processes.

Construction and interpretation: Counting processes are commonly constructed from point processes, with N(t) defined as the number of event occurrences in the time interval (0, t]. They are used to model arrivals, events, or incidents in time, including systems like queues, reliability models, neural spike trains, and finance-related event data.

Related concepts include stochastic intensity or compensator, which describe the instantaneous rate of events, and multivariate or marked counting processes, which extend the idea to multiple event types or additional event information.