# Nonhomogeneous Poisson and Heavy tail inter arrival time distribution

What is the relationship between a Nonhomogeneous Poisson process and a process that has heavy tail distribution for its inter arrival times?

Any pointer to a resource that can shed some light on this question would be hugely appreciated

Well, if you have a point process that you try modeling as a Poisson process, and find it has heavy tails, there are several possibilities. What are the key assumptions for a Poisson Process:

-There is a constant rate function -Events are memoryless, that is P(E in (t,t+d)) is independent of t and when other events are. -The waiting time until the next event is exponentially distributed (kinda what the previous two are saying)

So, how can you violate these assumptions to get heavy tails?

-Non-constant rate function. If the rate function switches between, say, two values, you'll have too many short wait-times, and too many long wait-times, given the overall rate function. This can show itself as having heavy tails. -The waiting time is not exponentially distributed. In which case, you don't have a Poisson process. You have some other sort of point process.

Note that in the extreme case, any point process can be modeled by a NHPP - put a delta function at each event, and set the rate to 0 elsewhere. I think we can all agree that this is a poor model, having little predictive power. So if you are interested in a NHPP, you'll want to think a bit about whether that is the right model, or whether you are overly-adjusting a model to fit your data.

In general there is none. A Poisson process has inter-arrival times that are exponentially distributed, which does not have heavy tails.

• That is only true for homogeneous Poisson Processes. If you have a Non-homogeneous Poisson process (that is not equivalent to some homogeneous Poisson process), then the inter-arrival times will not be exponentially distributed. – mpacer Jan 13 '15 at 1:00

The assumption that

Heavy tail distribution for its inter arrival times

along with the assumption that each inter-arrival time is independent, you have a renewal process. There are different ways that a renewal process can have a inhomogeneous marginal rate. Most frequently this is done by stretch time (time-rescaling theorem), or scale the hazard function (multiplication of the rate and the hazard function at each time point). Poisson process is a special case where the inter-arrival time is exponential as user549 answered. However, making inhomogeneous Possion with either method results in the same inter-spike interval distribution.

In summary, inhomogeneous Poisson process cannot have heavy tails.