Some signals have no waveform. Consider the measurement of when lightning strikes occur within some region; the random process is 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 function consists 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 t 1 t 2 . Consequently, N t N 0 , t . The event 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 quantity known as the intensity λ ( t ; N t ; W ) of the point process through the probability law N t ; W N t , t + Δ t 1 λ ( t ; N t ; W ) Δ t for sufficiently small Δ t . Note that this probability is a conditional probability; it can depend on how many events occurred previously and when they occurred. The intensity can also vary with time to describe non-stationary point processes. The intensity 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. A stationary Poisson process results when the intensity equals a constant: λ ( t ; N t ; W ) λ 0 . Thus, in a Poisson process, a coin is flipped every Δ t seconds, with a constant probability of heads (an event) occuring that equals λ 0 Δ t and is independent of the occurrence of past (and future) events. When this probability varies with time, the intensity equals λ t , a non-negative signal, and a nonstationary Poison process results. In the literature, stationary Poisson processes are sometimes termed homogeneous, nonstationary ones inhomogeneous. From the Poisson process's definition, we can derive the probability laws that govern event occurrence. These fall into two categories: the count statistics N t 1 , t 2 n , the probability of obtaining n events in an interval t 1 t 2 , and the time of occurrence statistics p W n w , the joint distribution of the first n event times in the observation interval. These times form the vector W n , 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 disjoint intervals. Let t 1 be fixed and consider event occurrence in the intervals t 1 t 2 and t 2 t 2 δ , and how these contribute to the occurrence of n events in the union of the two intervals. If k events occur in t 1 t 2 , then n k must occur in t 2 t 2 δ . Furthermore, the scenarios for different values of k are mutually exclusive. Consequently, N t 1 , t 2 + δ n k 0 n N t 1 , t 2 k N t 2 , t 2 + δ n k N t 1 , t 2 n N t 2 , t 2 + δ 0 N t 1 , t 2 n N t 1 , t 2 n 1 N t 2 , t 2 + δ 1 N t 1 , t 2 n 1 k 2 n N t 1 , t 2 n k N t 2 , t 2 + δ k N t 1 , t 2 n k Because of the independence of event occurrence in disjoint intervals, the conditional probabilities in this expression equal the unconditional ones. When δ is small, only the first two will be significant to the first order in δ. Rearranging and taking the obvious limit, we have the equation defining the count statistics. t 2 N t 1 , t 2 n λ t 2 N t 1 , t 2 n λ t 2 N t 1 , t 2 n 1 To solve this equation, we apply a z-transform to both sides. Defining the transform of N t 1 , t 2 n to be P t 2 z , Remember, t 1 is fixed and can be suppressed notationally. we have t 2 P t 2 z λ t 2 1 z -1 P t 2 z Applying the boundary condition that P t 1 z 1 , this simple first-order differential equation has the solution P t 2 z 1 z -1 α t 1 t 2 λ α To evaluate the inverse z-transform, we simply exploit the Taylor series expression for the exponential, and we find that a Poisson probability mass function governs the count statistics for a Poisson process. N t 1 , t 2 n α t 1 t 2 λ α n n α t 1 t 2 λ α The integral of the intensity occurs frequently, and we succinctly denote it by Λ t 1 t 2 . When the Poisson process is stationary, the intensity equals a constant, and the count statistics depend only on the difference t 2 t 1 .

Time of occurrence statistics To derive the multivariate distribution of W, we use the count statistics and the independence properties of the Poisson process. The density we seek satisfies v w 1 w 1 δ 1 … v w n w n δ n p W n v W 1 w 1 w 1 δ 1 … W n w n w n δ n The expression on the right equals the probability that no events occur in t 1 w 1 , one event in w 1 w 1 δ 1 , no event in w 1 δ 1 w 2 , etc. Because of the independence of event occurrence in these disjoint intervals, we can multiply together the probability of these event occurrences, each of which is given by the count statistics. W 1 w 1 w 1 δ 1 … W n w n w n δ n Λ t 1 w 1 Λ w 1 w 1 + δ 1 Λ w 1 w 1 + δ 1 Λ w 1 + δ 1 w 2 Λ w 2 w 2 + δ 2 Λ w 2 w 2 + δ 2 … Λ w n w n + δ n Λ w n w n + δ n k 1 n λ w k δ k Λ t 1 w n for small δ k . From this approximation, we find that the joint distribution of the first n event times equals p W n w k 1 n λ w k α t 1 w n λ α t 1 w 1 w 2 … w n 0

Sample function density For Poisson processes, the sample function density describes the joint distribution of counts and event times within a specified time interval. Thus, it can be written as t 1 t t 2 N t W 1 w 1 … W n w n N t 1 , t 2 n p W n w The second term in the product equals the distribution derived previously for the time of occurrence statistics. The conditional probability equals the probability that no events occur between w n and t 2 ; from the Poisson process's count statistics, this probability equals Λ w n t 2 . Consequently, the sample function density for the Poisson process, be it stationary or not, equals t 1 t t 2 N t k 1 n λ w k α t 1 t 2 λ α

Properties From the probability distributions derived on the previous pages, we can discern many structural properties of the Poisson process. These properties set the stage for delineating other point processes from the Poisson. They, as described subsequently, have much more structure and are much more difficult to handle analytically.

The Counting Process The counting process N t is an independent increment process. For a Poisson process, the number of events in disjoint intervals are statistically independent of each other, meaning that we have an independent increment process. When the Poisson process is stationary, increments taken over equi-duration intervals are identically distributed as well as being statistically independent. Two important results obtain from this property. First, the counting process's covariance function K N t u equals σ 2 t u . This close relation to the Wiener waveform process indicates the fundamental nature of the Poisson process in the world of point processes. Note, however, that the Poisson counting process is not continuous almost surely. Second, the sequence of counts forms an ergodic process, meaning we can estimate the intensity parameter from observations. The mean and variance of the number of events in an interval can be easily calculated from the Poisson distribution. Alternatively, we can calculate the characteristic function and evaluate its derivatives. The characteristic function of an increment equals Φ N t 1 , t 2 v v 1 Λ t 1 t 2 The first two moments and variance of an increment of the Poisson process, be it stationary or not, equal N t 1 , t 2 Λ t 1 t 2 N t 1 , t 2 2 Λ t 1 t 2 Λ t 1 t 2 2 N t 1 , t 2 Λ t 1 t 2 Note that the mean equals the variance here, a trademark of the Poisson process.

Poisson process event times from a Markov process Consider the conditional density p W n | W n - 1 , … , W 1 w n | w n - 1 , … , w 1 . This density equals the ratio of the event time densities for the n- and ( n 1 )-dimensional event time vectors. Simple substitution yields w n w n w n - 1 p W n | W n - 1 , … , W 1 w n | w n - 1 , … , w 1 λ w n α w n - 1 w n λ α Thus the n th event time depends only on when the n 1 th event occurs, meaning that we have a Markov process. Note that event times are ordered: the n th event must occur after the n 1 th , etc. Thus, the values of this Markov process keep increasing, meaning that from this viewpoint, the event times form a nonstationary Markovian sequence. When the process is stationary, the evolutionary density is exponential. It is this special form of event occurence time density that defines a Poisson process.

Interevent intervals in a Poisson process form a white sequence. Exploiting the previous property, the duration of the n th interval τ n w n w n - 1 does not depend on the lengths of previous (or future) intervals. Consequently, the sequence of interevent intervals forms a "white" sequence. The sequence may not be identically distributed unless the process is stationary. In the stationary case, interevent intervals are truly white - they form an IID sequence - and have an exponential distribution. τ τ 0 p τ n τ λ 0 λ 0 τ To show that the exponential density for a white sequence corresponds to the most "random" distribution, Parzen proved that the ordered times of n events sprinkled independently and uniformly over a given interval form a stationary Poisson process. If the density of event sprinkling is not uniform, the resulting ordered times constitute a nonstationary Poisson process with an intensity proportional to the sprinkling density.

Doubly stochastic Poisson processes Here, the intensity λ t equals a sample function drawn from some waveform process. In waveform processes, the analogous concept does not have nearly the impact it does here. Because intensity waveforms must be non-negative, the intensity process must be nonzero mean and non-Gaussian. The authors shall assume throughout that the intensity process is stationary for simplicity. This model arises in those situations in which the event occurrence rate clearly varies unpredictably with time. Such processes have the property that the variance-to-mean ratio of the number of events in any interval exceeds one. In the process of deriving this last property, we illustrate the typical way of analyzing doubly stochastic processes: Condition on the intensity equaling a particular sample function, use the statistical characteristics of nonstationary Poisson processes, then "average" with respect to the intensity process. To calculate the expected number N t 1 , t 2 of events in an interval, we use conditional expected values: N t 1 , t 2 λ t t 1 t t 2 N t 1 , t 2 α t 1 t 2 λ α t 2 t 1 λ t This result can also be written as the expected value of the integrated intensity: N t 1 , t 2 Λ t 1 t 2 . Similar calculations yield the increment's second moment and variance. N t 1 , t 2 2 Λ t 1 t 2 Λ t 1 t 2 2 N t 1 , t 2 Λ t 1 t 2 Λ t 1 t 2 Using the last result, we find that the variance-to-mean ratio in a doubly stochastic process always exceeds unity, equaling one plus the variance-to-mean ratio of the intensity process. The approach of sample-function conditioning can also be used to derive the density of the number of events occurring in an interval for a doubly stochastic Poisson process. Conditioned on the occurrence of a sample function, the probability of n events occurring in the interval t 1 t 2 equals () λ t t 1 t t 2 N t 1 , t 2 n Λ t 1 t 2 n n Λ t 1 t 2 Because Λ t 1 t 2 is a random variable, the unconditional distribution equals this conditional probability averaged with respect to this random variable's density. This average is known as the Poisson Transform of the random variable's density. N t 1 , t 2 n α 0 α n n α p Λ t 1 t 2 α