The Poisson Counting Process as an Example

Counting Arrivals Over Continuous Time

A counting process records how many events have occurred by each time. Unlike a sequence that records one new observation at each integer time, a continuous-time counting process can change at any nonnegative real time. The Poisson counting process is a basic example: its values are counts, its paths move only upward, and the probability law of its observations can be calculated from the lengths of the time intervals.

Fix a rate \(\lambda>0\) and a probability space \((\Omega,\mathcal{F},\mathbb{P})\). One construction starts with independent interarrival times \(\tau_1,\tau_2,\ldots\), each having the exponential distribution with rate \(\lambda\), so that \(\mathbb{P}(\tau_k>x)=e^{-\lambda x}\) for \(x\ge0\). Define the arrival times \(S_k=\tau_1+\cdots+\tau_k\). The count at time \(t\) is the number of arrival times at or before \(t\).

The process is continuous in its time index, but it does not take arbitrary real values. At each fixed time its value is a nonnegative integer. Its sample path is a step function: it stays at its current count between arrivals and increases by one at each arrival. The definition uses \(\le t\), so the value at an arrival time is the value after the jump.

The arrival construction does not produce infinitely many arrivals in a finite interval. Here is a direct check. Each event \(\{\tau_k\ge 1/\lambda\}\) has probability \(e^{-1}>0\), and these events are independent. For any fixed integer \(m\), the probability that none of the next \(r\) interarrival times is at least \(1/\lambda\) is \((1-e^{-1})^r\), which tends to zero as \(r\) tends to infinity. Thus, with probability one, there are infinitely many such interarrival times after every \(m\). Taking the countable intersection over \(m\), their sum diverges, so \(S_k\to\infty\) almost surely. In particular, only finitely many arrivals occur by any fixed finite time.

Path Properties

Proof. Since every \(\tau_k\) is positive almost surely, \(S_k>0\), and hence no arrival is counted at time zero: \(N_0=0\). For any finite \(t\), the divergence \(S_k\to\infty\) means that only finitely many \(k\) satisfy \(S_k\le t\); therefore the defining sum is a finite sum of indicators, and \(N_t\) is a nonnegative integer. If \(s\le t\), then every arrival counted by time \(s\) is also counted by time \(t\), so \(N_s\le N_t\). This proves nondecreasing paths. On any bounded time interval there are only finitely many arrivals. At an arrival time \(S_k\), the convention \(S_k\le t\) includes that arrival at \(t=S_k\), and \(N_t=k\) until the next arrival. Away from arrivals the count is constant in some interval immediately to the right. Thus the path is right-continuous at every time. The finite number of arrivals on bounded intervals also gives a left-hand limit at every positive time. Since the arrival times are strictly increasing, the count increases by exactly one at each \(S_k\). \(\square\)

These are pathwise properties: they describe what the process looks like for almost every outcome. They are different from statements about the distribution of the count at a fixed time. The latter can be derived from the interarrival construction.

Proof. For \(t>0\), the event \(N_t=k\) is the event \(S_k\le t<S_{k+1}\) when \(k\ge1\). The density of \(S_k\) is \(f_{S_k}(s)=\lambda^k s^{k-1}e^{-\lambda s}/(k-1)!\) for \(s>0\). To obtain this, start with the exponential density for \(k=1\); convolving the density for \(S_k\) with \(\lambda e^{-\lambda x}\) gives \(\int_0^s [\lambda^k u^{k-1}e^{-\lambda u}/(k-1)!]\lambda e^{-\lambda(s-u)}\,du=\lambda^{k+1}s^ke^{-\lambda s}/k!\), which is the claimed form for \(k+1\).

Given \(S_k=s\le t\), the next interarrival time is independent of \(S_k\), and the probability it exceeds \(t-s\) is \(e^{-\lambda(t-s)}\). Consequently, for \(k\ge1\), \(\mathbb{P}(N_t=k)=\int_0^t f_{S_k}(s)e^{-\lambda(t-s)}\,ds=e^{-\lambda t}\lambda^k t^k/k!\), since \(\int_0^t s^{k-1}\,ds=t^k/k\). For \(k=0\), the event is \(\tau_1>t\), whose probability is \(e^{-\lambda t}\). This proves the formula for every \(k\in S\). \(\square\)

The exponential waiting-time construction also explains why counts over disjoint time intervals are independent. The key identity is the memoryless property: for \(x,y\ge0\), \(\mathbb{P}(\tau_k>x+y\mid\tau_k>x)=e^{-\lambda y}\). At a fixed time \(s\), conditional on the arrival times observed by \(s\), the waiting time remaining until the next arrival therefore has the same exponential distribution as a fresh interarrival time; subsequent interarrival times are independent of the observed ones. Thus the arrivals after \(s\) have the same count law as a fresh process and are independent of the arrival history through \(s\). Applying this successively at interval endpoints gives independent counts on disjoint intervals. In particular, \(N_t-N_s\) has the Poisson distribution with mean \(\lambda(t-s)\) for \(0\le s<t\).

Observations at Several Times

A finite-dimensional observation is the vector of process values at finitely many specified times. If \(0\le t_1<\cdots<t_m\), the vector \((N_{t_1},\ldots,N_{t_m})\) takes values in \(S^m\), but its coordinates are ordered: every possible observation satisfies \(N_{t_1}\le\cdots\le N_{t_m}\). It is often simpler to calculate probabilities for the increments first and then recover the counts by adding them.

Proof. Define the increments \(Y_r=N_{t_r}-N_{t_{r-1}}\), with \(N_{t_0}=N_0=0\). The event in the formula is exactly the event \(Y_r=n_r-n_{r-1}\) for every \(r\): summing the increments from \(1\) through \(r\) recovers \(N_{t_r}=n_r\). Counts on the disjoint intervals \((t_{r-1},t_r]\) are independent, and each has the Poisson distribution with mean \(\lambda(t_r-t_{r-1})\). Multiplying their probabilities gives the displayed product. Each increment must be nonnegative, so if some \(n_r<n_{r-1}\), the event is impossible. \(\square\)

What the Count Process Does—and Does Not—Say

A useful way to read a Poisson counting process is as a random, nondecreasing integer-valued function of continuous time. The finite-dimensional distribution describes probabilities for selected observation times; it is not itself a sample path. Conversely, path properties such as monotonicity do not specify the probability law. Both kinds of information matter: the path tells how a realized count evolves, while the distribution tells how likely different observations are.

Common pitfall: excluding an arrival at the endpoint. Suppose one arrival occurs at time \(S_1=\tfrac34\). The incorrect calculation \(N_{3/4}=0\) treats the count as arrivals strictly before \(\tfrac34\). The definition instead uses \(S_k\le t\), so \(N_{3/4}=1\). Correspondingly, the number of arrivals in \((s,t]\) is \(N_t-N_s\); changing the endpoint convention can change which interval receives an arrival exactly at a boundary. In the exponential construction an arrival at a specified deterministic time has probability zero, but the pathwise definition still needs a consistent endpoint rule.

Another mistake is to treat the coordinates \(N_{t_1},N_{t_2}\) as independent. They are not: when \(t_1\le t_2\), the first count is included in the second, and \(N_{t_2}\ge N_{t_1}\) on every path. Independence applies to counts on disjoint intervals, such as \(N_{t_1}\) and \(N_{t_2}-N_{t_1}\), not generally to the cumulative counts themselves.