1.9 The poisson process

Some signals have no waveform. Consider the measurement of when lightning strikes occur within some region; the random processis the sequence of event times, which has no intrinsic waveform. Such processes are termed point processes , and have been shown (see Snyder ) to have simple mathematical structure. Define some quantities first.Let $N_t$ be the number of events that have occurred up to time $t$ (observations are by convention assumed to start at $t=0$ ). This quantity is termed the counting process, and has the shape of a staircase function: The counting functionconsists of a series of plateaus always equal to an integer, with jumps between plateaus occuring when events occur.The increment $N_{t_1,t_2}=N_{t_2}-N_{t_1}$ corresponds to the number of events in the interval $\left[t_1 , t_2\right)$ . Consequently, $N_t=N_{0,t}$ . Theevent times comprise the random vector $W$ ; the dimension of this vector is $N_t$ , the number of events that have occured. The occurrence of events is governed by a quantityknown as the intensity $\left(t;N_t;\mathbf{W}\right)$ of the point process through the probability law $$(N_t;\mathbf{W}, N_{t,t+t}=1)=\left(t;N_t;\mathbf{W}\right)(t)$$ for sufficiently small $(t)$ . Note that this probability is a conditional probability; it can depend on how many events occurredpreviously and when they occurred. The intensity can also vary with time to describe non-stationary point processes. Theintensity has units of events, and it can be viewed as the instantaneous rate at which events occur.

The simplest point process from a structural viewpoint, the Poisson process, has no dependence on process history. Astationary Poisson process results when the intensity equals a constant: $\left(t;N_t;\mathbf{W}\right)={}_0$ . Thus, in a Poisson process, a coin is flipped every $(t)$ seconds, with a constant probability of heads (an event) occuring that equals ${}_0(t)$ and is independent of the occurrence of past (and future) events. When this probability varies with time, theintensity equals $(t)$ , a non-negative signal, and a nonstationary Poison process results.

From the Poisson process's definition, we can derive the probability laws that govern event occurrence. These fall intotwo categories: the count statistics $(N_{t_1,t_2}=n)$ , the probability of obtaining $n$ events in an interval $\left[t_1 , t_2\right)$ , and the time of occurrence statistics $p({\mathbf{W}}^{\left(n\right)}, w)$ , the joint distribution of the first $n$ event times in the observation interval. These times form the vector $W^{\left(n\right)}$ , the occurrence time vector of dimension $n$ . From these two probability distributions, we can derive the sample function density.

Count statistics

We derive a differentio-difference equation that $(N_{t_1,t_2}=n)$ , $t_1< t_2$ , must satisfy for event occurrence in an interval to be regular and independent of event occurrences in disjointintervals. Let $t_1$ be fixed and consider event occurrence in the intervals $\left[t_1 , t_2\right)$ and $\left[t_2 , t_2+\right)$ , and how these contribute to the occurrence of $n$ events in the union of the two intervals. If $k$ events occur in $\left[t_1 , t_2\right)$ , then $n-k$ must occur in $\left[t_2 , t_2+\right)$ . Furthermore, the scenarios for different values of $k$ are mutually exclusive. Consequently,