# Relationship between Poisson distribution and Poisson Process

• A
SunilS
TL;DR Summary
How bijective is this mapping?
Apologies if this has been discussed elsewhere.

I know a Poisson process implies a Poisson distribution, but does a Poisson distribution imply a Poisson process? and does the absence of a Poisson distribution imply the absence of a Poisson process?

TIA - Sunil

Gold Member
I know a Poisson process implies a Poisson distribution... and does the absence of a Poisson distribution imply the absence of a Poisson process?

this seems like basic logic you can handle...

more interesting is your question is does poisson distribution imply poisson process. I'll state an answer of no. You may be able to torture the common sense examples that follow, but I don't see it.

Do you have a definition of poisson process? One is an idealized renewal process with exponential inter-arrival times. What does this have to do with things like birthday problems or derangments problems? They aren't really renewal problems in my book and hence not really poisson processes.

The poisson distribution -- either in as an exact distribution for a problem or as a very close approximation (reference stein-chen if you'd like) is extremely handy even when completely divorced from renewal theory.

SunilS
I know that a Poisson distribution does not imply a Poisson process, but it is not obvious to me that the inverse mapping works the way I think it does.

That's why I am looking for an explicit discussion. I think there is one in Kingman's book but I don't have access right now (working on getting it).

I wasn't thinking of the Poisson Process in terms of inter-arrival times but rather just independence between consecutive events.

I am not familiar with the literature on this, and couldn't find anything on Arxiv that helped quickly - so any pointers would be helpful.

SunilS
Thank you - I will look at this. The Kingman book should arrive in a few days, so hopefully I will have more insight at that point also.

SunilS
Actually I just realized that there is a simpler way of posing my problem.

I am measuring what should be uncorrelated events but I observe a non-Poisson distribution.

Does that mean

a) My system has not thermalized (become memoryless/renewed) between observation times? or

b) There is a hidden source of random noise in my detection that is convolving in a way that effectively de-thermalizes the combination of the system under observation and the measurement apparatus?

And how does one tell one from the other - given that I do not have a good model for my detector and the noise I should see in it.

Gold Member
Actually I just realized that there is a simpler way of posing my problem.

I am measuring what should be uncorrelated events but I observe a non-Poisson distribution.

Does that mean

a) My system has not thermalized (become memoryless/renewed) between observation times? or

b) There is a hidden source of random noise in my detection that is convolving in a way that effectively de-thermalizes the combination of the system under observation and the measurement apparatus?

And how does one tell one from the other - given that I do not have a good model for my detector and the noise I should see in it.
This feels like a new thread. What you've stated is rather problematic. Thermalized is not a standard term at all in probability.

You haven't described the (believed) generator of these events, so a statement like
I am measuring what should be uncorrelated events but I observe a non-Poisson distribution.

I don't understand why you used the term 'uncorrelated' here. Do you mean independent? As clear from an adjacent thread going on right now, an awful lot of people don't understand the meaning of uncorrelated.

Even if the events are independent, what does this have to do with being a Poisson process-- there many partial sums of iid non-negative random variables that are renewal processes... the (homogenous) poisson process is a very special case where the iid non-negative random variables are exponentially distributed.

SunilS
Yes it is a different thread - I want to avoid proliferation at this time .

To me "uncorrelated" means "independent" in the physical sense as in I have some equations that could be used to describe my system but its interactions with a "thermal bath" destroy the determinate nature of the system observables. I end up with a certain randomness in the observable system behavior. The observed system behavior at time t_n should be uncorrelated with the observed behavior at t_(n-1) and t_(n+1).

I realize that is not what everyone else does. I realize there is an established vocabulary in the probability field and I don't want to get into a semantic discussion right now

That part about "many partial sums of iid non-negative random variables" is what I am trying to understand. That's the part I am unfamiliar with.