Stationary Increments Versus Stationary Processes

Two Different Meanings of “Stationary”

A Poisson process has a time-homogeneous pattern of arrivals: the number of arrivals in an interval depends on the interval's length, not on where it starts. Yet the process itself records the cumulative number of arrivals since time zero, so its values generally change in distribution as time passes. These are different statements. Separating them matters whenever data are described as “stationary”: it is important to know whether that means counts on shifted windows, or the joint laws of the process values themselves.

Work on a probability space \((\Omega,\mathcal{F},\mathbb{P})\). Let \((N_t)_{t\ge0}\) be a Poisson process with rate \(\lambda\ge0\), taking values in the state space \(S=\{0,1,2,\ldots\}\). Thus \(N_0=0\), its increments over disjoint time intervals are independent, and for \(0\le a<b\), \(N_b-N_a\) has a Poisson distribution with mean \(\lambda(b-a)\). In particular, for \(r=0,1,2,\ldots\),

When \(\lambda=0\), this formula is interpreted as a point mass at zero. The count \(N_b-N_a\) is the number of arrivals in the interval \((a,b]\). The notation makes the distinction central to this tutorial: \(N_t\) is a process level, while \(N_b-N_a\) is an interval count.

Stationary increments compare differences, whereas strict stationarity compares levels. In the strict-stationarity definition, shifting all observation times also shifts the amount of history accumulated by each observation time. For a process that starts at zero, those levels need not have the same laws.

Shifted Interval Counts

The Poisson count formula gives stationary increments directly. An interval of length \(t\) has a Poisson count with mean \(\lambda t\), regardless of its starting time \(h\). Moreover, independent increments give a joint statement for multiple intervals: shifting a collection of disjoint intervals by the same amount preserves both their lengths and the independence of their counts.

Proof. Set \(\ell_j=t_j-t_{j-1}\), which is positive for each \(j\). For either collection of intervals, the \(j\)th count has a Poisson distribution with mean \(\lambda\ell_j\). The intervals in each collection are disjoint, so the \(m\) counts are independent. For any nonnegative integers \(r_1,\ldots,r_m\), the probability that the vector equals \((r_1,\ldots,r_m)\), for either collection, is therefore \[ \prod_{j=1}^m e^{-\lambda\ell_j}\frac{(\lambda\ell_j)^{r_j}}{r_j!}, \] with the zero-rate interpretation given above. These point probabilities agree for every vector of possible count values, proving equality of the joint distributions and hence stationary increments. Now suppose \(\lambda>0\). Strict stationarity would require \(N_1\) and \(N_2\) to have the same distribution, by choosing one observation time \(t_1=1\) and shift \(h=1\). But \(\mathbb{P}(N_1=0)=e^{-\lambda}\), whereas \(\mathbb{P}(N_2=0)=e^{-2\lambda}\); these differ because \(\lambda>0\). Thus strict stationarity fails. If \(\lambda=0\), every increment is zero almost surely, and \(N_t=0\) almost surely for every \(t\). Every finite vector of levels is then the zero vector, so strict stationarity holds. \(\square\)

Why the Process Levels Are Not Stationary

At time \(t\), \(N_t=N_t-N_0\) counts all arrivals since the process began. Its distribution is Poisson with mean \(\lambda t\). As \(t\) increases, that mean increases when \(\lambda>0\), so the one-time distributions change. This already rules out strict stationarity, which requires unchanged distributions after every common time shift.

The difference also appears when comparing joint distributions at shifted observation times. For \(0\le s<t\), the levels \(N_s\) and \(N_t\) record the count by time \(s\) and the later count by time \(t\). Their relationship is governed by the increment \(N_t-N_s\), but their joint law includes the distribution of the earlier level \(N_s\). Shifting both times changes the amount of time from zero to the first observation.

This joint comparison is consistent with stationary increments. In the two pairs, the increment from the first observation time to the second has the same distribution: \(N_2-N_1\) and \(N_3-N_2\) are both Poisson with mean \(1\). But the first levels, \(N_1\) and \(N_2\), do not have the same distribution. Equality of the increment laws does not make the corresponding vectors of levels equal in distribution.

A Common Mistake and Its Correction

A frequent error is to reason: “The process has stationary increments, so \(N_{t+h}\) has the same distribution as \(N_t\).” The stationary-increments statement is instead that \(N_{t+h}-N_h\) has the same distribution as \(N_t-N_0\). It concerns the count accumulated during a length-\(t\) window, not the count accumulated from time zero to the end of that window.

For a concrete correction, let \(\lambda=1\), \(t=1\), and \(h=1\). The incorrect comparison would equate \(N_2\) and \(N_1\). But \(\mathbb{P}(N_2=0)=e^{-2}\) and \(\mathbb{P}(N_1=0)=e^{-1}\), so those levels do not have the same distribution. The valid stationary-increments comparison is between \(N_2-N_1\) and \(N_1-N_0\). Both count arrivals in an interval of length \(1\), and each has probability \(e^{-1}\) of being zero.

Keep the following distinctions in view:

• Stationary increments: shifting an interval or a collection of intervals leaves the joint law of their counts unchanged.
• Strict stationarity: shifting observation times leaves every finite-dimensional joint law of process levels unchanged.
• For positive-rate Poisson processes: stationary increments hold, but strict stationarity fails.
• For zero-rate Poisson processes: all counts are zero, so both properties hold.

These terms describe different features of a model. When examining observed data, counts over equal-length windows can be identically distributed even while the cumulative count has a changing distribution. Conversely, checking only one-time interval-count distributions does not establish the joint behavior of counts on multiple windows; independence is an additional part of the Poisson-process structure.