Counting Process
A counting process is a stochastic process whose value represents the number of events that have occurred by a specified time. For a process ({N(t):t\geq 0}), the standard definition requires that (N(0)=0), that every (N(t)) is a nonnegative integer, and that each sample path is nondecreasing and right-continuous. A counting process is called simple when every jump has size one, so simultaneous events are excluded.
Counting processes provide a common mathematical representation for arrivals, failures, transactions, and other events localized in time. Their probabilistic structure depends not only on the distribution of the total number of events but also on the dependence between event times. The Poisson process is the principal model with independent increments, while renewal processes and processes with stochastic intensities describe broader forms of temporal dependence.
Event times and sample paths
For a simple counting process, the (n)-th event time is
[ T_n=\inf{t\geq 0:N(t)\geq n}. ]
The counting process can then be reconstructed from its event times through
[ N(t)=\sum_{n\geq 1}\mathbf{1}_{{T_n\leq t}}, ]
where (\mathbf{1}) denotes an indicator function. Conversely, the ordered jump times of a non-explosive simple counting process determine its entire sample path. Non-explosion means that only finitely many events occur in every bounded time interval, or equivalently that (T_n) tends to infinity as (n) increases.
The waiting times
[ X_n=T_n-T_{n-1},\qquad T_0=0, ]
provide an alternative description. Their joint distribution determines whether the process has independent increments, whether the event rate changes with time, and whether previous arrivals influence future ones.
A nonsimple counting process permits jumps larger than one. Such a jump records several events at the same instant and is naturally represented by an integer-valued random measure. If (\mu) is the associated measure on the time axis, then
[ N(t)=\mu((0,t]). ]
This formulation extends directly to marked point processes, in which every event carries an additional random quantity such as a spatial position or event magnitude.
Poisson counting processes
A homogeneous Poisson process with rate (\lambda>0) is a simple counting process satisfying
[ N(t)-N(s)\sim \operatorname{Poisson}(\lambda(t-s)) ]
for (0\leq s<t), with increments over disjoint intervals being independent. Consequently,
[ \Pr(N(t)=n)=e^{-\lambda t}\frac{(\lambda t)^n}{n!}. ]
Its interarrival times are independent exponential random variables with mean (1/\lambda). The memoryless property of the exponential distribution corresponds to the absence of duration dependence in the arrival mechanism.
Siméon Denis Poisson established the distribution that later supplied the one-dimensional count law, while A. K. Erlang created early arrival-process methods for telephone traffic. The subsequent measure-theoretic formulation separated the count distribution from the stronger requirement that increments over disjoint intervals be independent.
For a nonhomogeneous Poisson process, the deterministic rate is a function (\lambda(t)), and the cumulative rate is
[ \Lambda(t)=\int_0^t\lambda(s),ds. ]
The increment (N(t)-N(s)) then has a Poisson distribution with mean (\Lambda(t)-\Lambda(s)). A deterministic change of clock by (\Lambda) converts the process into a homogeneous Poisson process whenever the cumulative rate is continuous and strictly increasing.
Filtrations, intensities, and compensators
A counting process is normally considered relative to a filtration ({\mathcal F_t}), representing the information available up to time (t). An intensity (\lambda_t) is a nonnegative predictable process that describes the instantaneous conditional event rate. Under the usual integrability conditions,
[ \Pr!\left(N(t+h)-N(t)=1\mid\mathcal F_t\right) =\lambda_t h+o(h), ]
while the conditional probability of multiple jumps during the same short interval is (o(h)).
The cumulative intensity
[ A_t=\int_0^t\lambda_s,ds ]
is the compensator when the intensity is absolutely continuous with respect to time. The compensated process
[ M_t=N(t)-A_t ]
is then a local martingale, and it is a martingale under appropriate integrability assumptions. This decomposition separates predictable accumulation from the innovation contributed by event occurrence.
Joseph L. Doob created the submartingale framework from which compensator theory developed, and Paul-André Meyer established the general decomposition now called the Doob–Meyer theorem. For an integrable counting process, the theorem produces a predictable increasing process (A) such that (N-A) is a martingale. The compensator need not possess a density, so an intensity of the form (\lambda_t) does not exist for every counting process.
Martingale characterization
In 1964, You Watanabe discovered the martingale characterization of the Poisson counting process. In its standard form, the result states that a non-explosive, adapted, simple counting process with (N(0)=0) is a Poisson process of rate (\lambda) when
[ N(t)-\lambda t ]
is a martingale relative to its filtration, subject to the filtration containing no anticipatory information about future jumps. The deterministic compensator (\lambda t) forces the conditional rate to remain constant and prevents past event times from modifying future arrival probabilities.
The corresponding time-inhomogeneous result replaces (\lambda t) with a deterministic continuous cumulative rate (\Lambda(t)). The characterization is stronger than an equality of expectations, since the condition
[ \mathbb E[N(t)]=\Lambda(t) ]
alone does not imply independent increments or Poisson-distributed counts. The martingale condition controls conditional increments and therefore identifies the process through its information structure rather than solely through its marginal distributions.
Renewal counting processes
A renewal process is obtained when the waiting times (X_1,X_2,\ldots) are independent and identically distributed nonnegative random variables. Its event times satisfy
[ T_n=X_1+\cdots+X_n, ]
and its count is
[ N(t)=\max{n:T_n\leq t}. ]
The Poisson process is the special renewal process whose waiting-time distribution is exponential. For other waiting-time laws, increments over disjoint intervals are generally dependent because the distribution of the next event depends on the time elapsed since the most recent renewal.
If the common waiting-time mean is finite and equal to (\mu), then the elementary renewal theorem gives
[ \lim_{t\to\infty}\frac{\mathbb E[N(t)]}{t}=\frac{1}{\mu}. ]
Under the corresponding pathwise assumptions, (N(t)/t) also converges to (1/\mu). This limit describes the long-run event rate without imposing the independent-increment structure of a Poisson process.
Random and history-dependent intensities
A Cox process has an intensity that is itself random. Conditional on the realized intensity path, the process is Poisson, but unconditional increments can be dependent because distinct time intervals share the same random environment. The model therefore separates event randomness from variation in the underlying rate.
A Hawkes process uses an intensity that depends on previous event times, commonly through
[ \lambda_t=\nu+\int_{(0,t)}g(t-s),dN(s), ]
where (\nu) is a baseline rate and (g) is a response kernel. A nonnegative kernel makes earlier events increase the subsequent conditional rate. Stability depends on the total influence of the kernel, with the standard stationary linear construction requiring
[ \int_0^\infty g(s),ds<1. ]
These processes differ from renewal models because their intensity may depend on the entire observed event history rather than only on the elapsed time since the most recent event.
Moments and dependence
For a counting process with integrable intensity,
[ \mathbb E[N(t)]=\mathbb E[A_t]. ]
When (N-A) is square-integrable and the process is simple, its predictable quadratic variation is determined by the compensator. For a homogeneous Poisson process,
[ \mathbb E[N(t)]=\operatorname{Var}(N(t))=\lambda t. ]
Equality between the mean and variance is therefore a consequence of the Poisson structure rather than a defining property of counting processes generally. Random intensities commonly produce variance exceeding the mean, while regular spacing between events can produce variance below it.
The covariance of Poisson counts satisfies
[ \operatorname{Cov}(N(s),N(t))=\lambda\min(s,t). ]
This expression follows from independent increments because the count at the earlier time is contained in the count at the later time. In history-dependent counting processes, the covariance additionally reflects interactions transmitted through the conditional intensity.