The Poisson Process – A Third Characteristion

There remains the matter of the distribution of the number of people to arrive in a fixed non-infinitissimal time interval. Consider the time interval [0,1], which we divide into n smaller intervals of equal width. As n grows large enough that we know the probability that two arrivals occur in the same interval tends to zero (as this is \leq no(\frac{1}{n})), we can consider this as a sequence of iid Bernoulli random variables as before. So

\mathbb{P}(N_1=k)=\binom{n}{k}(\frac{\lambda}{n})^k(1-\frac{\lambda}{n})^{n-k}
\approx \frac{n^k}{k!} \frac{\lambda^k}{n^k}(1-\frac{\lambda}{n})^n\approx \frac{\lambda^k}{k!}e^{-\lambda}.

We recognise this as belonging to a Poisson (hence the name of the process!) random variable. We can repeat this for a general time interval and obtain N_t\sim \text{Po}(\lambda t).

Note that we implicitly assumed that, in the infinitissimal case at least, behaviour in disjoint intervals was independent. We would hope that this would lift immediately to the large intervals, but it is not immediately obvious how to make this work. This property of independent increments is one of the key definitions of a Levy Process, of which the Poisson process is one of the two canonical examples (the other is Brownian Motion).

As before, if we can show that the implication goes both ways (and for this case it is not hard – letting t\rightarrow 0 clearly gives the infinitissimal construction), we can prove results about Poisson random variables with ease, for example \text{Po}(\lambda)+\text{Po}(\mu)=\text{Po}(\lambda+\mu).

This pretty much concludes the construction of the Poisson process. We have three characterisations:
1) X_n\sim\text{Exp}(\lambda) all iid.
2) The infinitissimal construction as before, with independence.
3) The number of arrivals in a time interval of width t \sim \text{Po}(\lambda t). (This is sometimes called a stationary increments property.) Furthermore, we have independent increments.

A formal derivation of the equivalence of these forms is important but technical, and so not really worth attempting here. See James Norris’s book for example for a fuller exposition.

The final remark is that the Poisson Process has the Markov property. Recall that this says that conditional on the present, future behaviour is independent of the past. Without getting into too much detail, we might like to prove this by using the independent increments property. But remember that for a continuous process, it is too much information to keep track of all the distributions at once. It is sufficient to track only the finite marginal distributions, provided the process is cadlag, which the Poisson process is, assuming we deal with the discontinuities in the right way. Alternatively, the exponential random variable is memoryless, a property that can be lifted, albeit with some technical difficulties, to show the Markov property.

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2 thoughts on “The Poisson Process – A Third Characteristion

  1. Pingback: Brownian Excursions and Local Time | Eventually Almost Everywhere

  2. Pingback: The Inspection Paradox and related topics | Eventually Almost Everywhere

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