# Defining a Poisson Process using Interarrival Times

The standard definition of the Poisson process with rate $\lambda$ is a stochastic counting process $\{N(t), t \geq 0\}$ such that

• $N(0) = 0$
• $\{N(t), t \geq 0\}$ has stationary and independent increments
• $\mathbb{P}[N(h) = 1] = \lambda h + o(h)$
• $\mathbb{P}[N(h) \geq 2] = o(h)$

Given the link between a Poisson process and the exponential distribution, I am wondering whether it is possible to define a Poisson process in terms of the interarrival times of events $\{T_n, n \geq 1\}$.

Here, $T_n$ is an exponentially distributed random variable with mean $\frac{1}{\lambda}$ that describes the time between the $n-1^\text{th}$ and $n^\text{th}$ arrival.

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Yes! Thanks for that. I've corrected it now. –  Elements Dec 11 '12 at 23:47
Specifically what are you looking to do? To map P(.. = ..) from N(t) into T? And are you familiar with the compound Poisson process? –  Arbias Hashani Dec 12 '12 at 1:09
Sorry I'm a little confused by your question. I am familiar with the Compound Poisson Process. I'm really just trying to find an alternative definition of the Poisson process that uses interarrival times in its definition instead of the number of arrivals in a given time interval. –  Elements Dec 12 '12 at 5:32

I am wondering whether it is possible to define a Poisson process in terms of the interarrival times of events $\{T_n, n \geq 1\}$.
This indeed can be done, and is done in most textbooks on the subject provided one adds to what you wrote the condition that the sequence $(T_n)_n$ is independent.