According to Ross[1], "the counting process $\{N(t), t \ge 0\}$ is said to be a Poisson process having rate $\lambda, \lambda>0$, if:
i) $N(0) = 0$.
ii) The process has independent increments.
iii) The number of events in any interval of length t is Poisson distributed with mean $\lambda t$. That is, for all $s, t \ge 0$,
$$ P\{N(t+s) - N(s) = n\} = e^{-\lambda t} \frac{(\lambda t)^n }{n!}$$ "
Is property (ii) needed in this definition? Doesn't property (iii) imply (ii)? If not, is there an example of a counting process that satisfies (i) and (iii) but not (ii)?
[1] Ross, S. M. (1983). Stochastic Processes. John Wiley & Sons.