# counting process, independent increments, stationarity

I am wondering if this definition of a counting process, implies some properties of the probability distribution associated with the counting process $\{N(t): t \ge 0\}$.

definition:

1. N(0)=0

2. The process is stationary and and has independent increments.

Does this definition imply:

1. The only parameter in the probability distribution for $N(t_{1})-N(t_{2})$ is $t_{1}-t_{2}=\Delta t$.

2. For any set of countable disjoint time increments: $\{[t^{1}_{1},t^{1}_{2}],[t^{2}_{1},t^{2}_{2}],[t^{3}_{1},t^{3}_{2}]........\}$, with lengths $\{\Delta t_{1},\Delta t_{2},\Delta t_{3}......\}$, and $\Delta T = \Sigma \Delta t_{i}$:

$P(N(\Delta T)=n) = \Sigma_{i,j,k....|i+j+k+...=n}[P(N(\Delta t_{1})=i)*P(N(\Delta t_{2})=j)*P(N(\Delta t_{3})=k)*.......]$

Do you agree with this implication?, and if so, is it and equivalence?

1. is not true. For a Poisson process with parameter $\lambda>0$ the distribution of your difference depends on $\lambda$ as well, so there are more parameters. But there are other processes taking values in $\mathbb{N}$ with stationary and independent increments.
2. Yes. This is true for processes with stationary and independent increments taking values in $\mathbb{N}$.
• Thank you for your help, I should have been clearer with 1 yes. Maybe one can say that it is only a function of a common parameter, andt $\Delta t$. Do you know if the property (2) has a name? I see that property 2 is sattisfied for a poisson distribution, but for other discrete distributions it may not be sattisfied. – user119615 Feb 22 '14 at 10:17