Poisson Processes as a Limit of Shrinking Bernoulli Processes (aka: pt.2)

So what exactly are Poisson processes useful for? Consider buses in a queue. Whenever a bus arrives a person enters the bus. The amount of time a person waits for the bus (that is, the amount of time before and up until an event occurs) is a random variable with some probability distribution. We can model bus arrivals through a sequence of these random variables,(which constitute their interarrival times). Specifically, we can approximate a Poisson process from this, provided our conditions fulfill various properties and definitions.

The previous post about Poisson processes was more or less describing how they behave, not really a definitive grasp of what they are based on the properties they have. By giving a set of definitions, one can construct a process that follows all definitions. This process is then described accordingly.

Definition: a Poisson counting process, N(t), is a counting process (that is, the aggregate number of arrivals by time t, where t > 0) that has the independent and stationary increment properties and fulfills the following properties:

Screen Shot 2015-05-04 at 7.14.53 PM

1) says that the probability of no arrivals is 1-λδ = 1-λ2j

2) says that the probability of exactly one arrival is λδ = λ2j

3) says that the probability of 2 or more arrivals is the higher order term upon expansion, which is essentially negligible.

Basically, what these properties are telling us are that there can be no more than one arrival at a time! The term 2j is really important, and we’ll see why in a little bit.

So how does this relate to a Bernoulli process? Well, Poisson processes are continuous time processes, and thus have an uncountably infinite set of random variables. This is particularly difficult to work with, so we approximate a continuous time process with a discrete time process. In this case, we can use a Bernoulli process, with some extra parameters, to approximate a Poisson process (as it turns out, the Bernoulli process is the correct approximation to use, and we’ll see why in a bit).

A Bernoulli process is an IID sequence of binary random variables i.e.; {X1, X2, X3, …, Xn} for which px(1) = p and px(0) = 1-p. So if a random variable has the value of 1, that is Xi = 1, there is a bus arrival at time i. If Xi = 0, there is no arrival at time i.

To make this process useful to approximate a Poisson process with, we “shrink” the time scale of the Bernoulli process s.t. there exists a j > 0, where Xi has either an arrival or no arrival at time i2-j.  This new time, i2-j, come from the fact that we are splitting the length of the interarrival period i by half, and then half, and so on, where each new slot has half the previous arrival probability.

To give a visual understanding of this, imagine that this is our original distribution timeline:

Screen Shot 2015-05-04 at 9.58.19 PM

For the first Bernoulli process (imagine there is a 1 as a superscript for each rv). Now, if we split the length of the interval in half,

Screen Shot 2015-05-04 at 10.02.38 PM

we get the 2nd Bernoulli process. This can go on and on, j times. This is done in such a way that the rate of arrivals per unit time is λ, thus Xi ~ Bernoulli(λ2-j).

So for each j, the “j-th” Bernoulli process has an associated counting process,

Screen Shot 2015-05-04 at 10.12.23 PM

Screen Shot 2015-05-04 at 10.16.12 PM

which is a binomial PMF.

Now we’re ready to tackle Poisson’s Theorem, which says:

Consider the sequence of shrinking Bernoulli processes with arrival probability λ2-j and time-slot size 2-j. Then for every fixed time t > 0 and fixed number of arrivals, n, the counting PMF approaches the Poisson PMF (of same λ) with increasing j.

That is to say,

Screen Shot 2015-05-04 at 10.23.29 PM

Proof:

Screen Shot 2015-05-04 at 10.36.39 PM

From the proof, we can see that a shrinking Bernoulli process is the appropriate approximation for a Poisson process, because the limit of the binomial PMF converges to the Poisson PMF.

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