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.