# Counting Process

Let Fig. 1 be a graph of the customers who are entering a bank. Every time a customer comes, the counter is increased by one. The time of the arrival of th customer is . Since the customers are coming at random, the sequence , denoted shortly by , is a random sequence. Also, the number of customers who came in the interval is a random variable (process). Such a process is right continuous, as indicated by the graph in Fig. 1.

As it is often case in the theory of stochastic processes, we assume that the index set, i.e. the set where is taking values from, is . Therefore, we have a sequence of non-negative random variables

as

WLOG1 let and , then

is called a point process (counting process), and is denoted shortly by .

Let be inter-arrival time, then the sequence of inter-arrival times is another stochastic process.

Special case is when is a sequence of i.i.d.2 random variables, then the sequence is called a renewal process. is the associated renewal point process, sometimes also called renewal process. Also, keep in mind that .

Definition (Poisson process) A point process is called a Poisson process if and satisfies the following conditions

1. its increments are stationary and its non-overlapping increments are independent
Remarks
• - is the increment of stochastic process .
• the number of new arrivals during .
• and is understood as when .

The Poisson process defined above is also known as homogeneous Poisson process. In general can be a time dependent function , in which case we are dealing with inhomogeneous Poisson process. Finally, itself can be a realization of stochastic process , in which case we have so-called doubly stochastic Poisson process.

In any case, the parameter of a Poisson process is called the rate and sometimes the intensity of the process. Its dimension is [events]/[time] (e.g. spikes/sec in neuroscience).

Theorem Let be a Poisson process, then

 (1)

The expression on the left hand side of (1) represents the probability of arrivals in the interval .

Proof A generating function of a discrete random variable is defined via the following z-transform (recall that the moment generating function of a continuous random variable is defined through Laplace transform):

where . Let us assume that is a Poisson random variable with parameter , then

and

 (2)

Going back to Poisson process, define the generating function as

Then we can write

Furthermore

Comparing this result to (2) we conclude that is a Poisson random variable with parameter .

Theorem If is a Poisson process and is the inter-arrival time between the th and th events, then is a sequence of i.i.d. random variables with exponential distribution, with parameter .

Proof

exponential

Need to show that and are independent and is also exponential.

 (3)

The event is a subset of the event described by Fig. 2, i.e.

no arrivals      one arrival         no arrivals

From (3)

 (4)

Similarly, the event described by Fig. 3 is a subset of the event , therefore

From (3)

 (5)

From (4) and (5), using squeeze theorem , it follows

Therefore, is independent of , and is exponentially distributed random variable.

Theorem

Proof Recall that , then

Likewise

Theorem (Conditioning on the number of arrivals) Given that in the interval the number of arrivals is , the arrival times are independent and uniformly distributed on .

Proof

Independence of arrival times , etc. directly follows from independence of non-overlapping increments. In particular let and be arrival times of first and second event, then

Suppose that we know exactly one event happened in the interval , and suppose the interval is partitioned into segments of length , as shown in Fig. 1. Let be the probability of event happening in the th  bin, then . From the definition of Poisson process it follows that , say . The constant is determined from

Let be a random variable corresponding to the time of arrival, then the probability density function (pdf) of can be defined as

 where

Therefore, is uniformly distributed on .
Let and be the arrival times of two events, and we know exactly two events happened on . Also assume that and represent mere labels of events, not necessarily their order. Given that happened in th bin, the probability of occurring in any bin of size is proportional to the size of that bin, i.e. , except for the th bin, where . By rendering the bin size infinitesimal, we notice that the probability remains constant over all but one bin, the bin in which occurred, where . But this set is a set of measure zero, so the cumulative sum over again gives rise to uniform distribution on .

Question What is the probability of observing events at instances on the interval ?

Since arrival times are continuous random variables, the answer is 0. However, we can calculate the associated pdf as

Question What is the power spectrum of Poisson process?

It does not make sense to talk about the power spectrum of Poisson process, since it is not a stationary process. In particular the mean of Poisson process is

and its autocorrelation function is

Since , we conclude that is not stationary (in weak sense), therefore it does not make sense to talk about its power spectrum. Let us define the following stochastic process (Fig. 5)

 spike train (6)

The fundamental lemma says that if , where is a linear operator, then

Since differentiation is a linear operator we have

Also, it can be shown using theory of linear operators that

Thus, is WWS3 stochastic process, and it makes sense to define the power spectrum of such a process as a Fourier transform of its autocorrelation function i.e.

Therefore, the spike train of independent times behaves almost as a white noise, since its power spectrum is flat for all frequencies, except for the spike at . The process defined by (6) is a simple version of what is in engineering literature known as a shot noise.

Definition (Inhomogeneous Poisson process) A Poisson process with a non-constant rate is called inhomogeneous Poisson process. In this case we have

1. non-overlapping increments are independent (the stationarity is lost though).

Theorem If is a Poisson process with the rate , then is a Poisson random variable with parameter i.e.

 (7)

Proof The proof of this theorem is identical to that of homogeneous case except that is replaced by . In particular, one can easily get

 (8)

Theorem Let be an inhomogeneous Poisson process with the rate and let , then

 (9)

The application of this theorem stems from the fact that we cannot use , since the increments are no longer stationary.

Proof

Thus, is a Poisson random variable with parameter , and (9) easily follows.

Theorem

Proof Recall that

 and

From (8) we have , and the two results follow after immediate calculations.

Theorem (Conditioning on the number of arrivals) Given that in the interval the number of arrivals is , the arrival times are independently distributed on with the pdf .

Proof The proof of this theorem is analogous to that of the homogeneous case. The probability of a single event happening at any of bins (Fig. 1) is given by , where is the bin index. Given that exactly one event occurred in the interval , we have

The argument for independence of two or more arrival times is identical to that of the homogeneous case.

Question What is the probability of observing events at instances on the interval ?

Since arrival times are continuous random variables, the answer is 0. However, we can calculate the associated pdf as

Question How to generate a sample path of a point (Poisson) process on ? It can be done in many different ways. For a homogeneous process we use three methods:

1. Conditioning on the number of arrivals. Draw the number of arrivals from a Poisson distribution with parameter , and then draw numbers uniformly distributed on .
2. Method of infinitesimal increments. Partition the segment into sufficiently small subsegments of length (say ). For each subsegment draw a number from a uniform distribution on . If , an event occurres in that subsegment. We repeat the procedure for all subsegments. We can use the approximation for a less expensive procedure.
3. Method of interarrival times. Using the fact that inter-arrival times are independent and exponentially distributed, we draw random variables from exponential distribution with parameter , where is the smallest integer that satisfies the criterion . Then the sequence generates the required point process.
For an inhomogeneous process, the procedures 1 and 2 can be modified using the appropriate theory of inhomegeneous Poisson process.

The illustration of the three procedures for a homogeneous case is given below. We use and . For each different procedure we generate sample paths, and calculate the statistics by averaging over ensemble. Raster plots of out of samples for the three methods are shown in Fig. 6, Fig. 7 and Fig. 8, respectively. Since the generated processes are homogeneous, the inter-arrival time distribution is exponential with parameter . The histograms of inter-arrival times corresponding to the three methods are also given below. They clearly exhibit an exponenital trend. The sample mean and sample standard deviation (unbiased) are also shown. Keep in mind that for exponentially distributed random variables, both mean and standard deviation are given by . Sample statistics indicates that is in the vicinity of .