Justification of Poisson postulates

Question

This may be a dumb question.

The Poisson postulates are:

1. $P(n=1,h) = \lambda h + o(h)$
2. $\sum\limits_{i=2}^{\infty}P(n=i,h) = o(h)$
3. Events in nonoverlapping intervals are independent

What ensures that $\lambda h \in [0,1]$ irrespective of the value of $\lambda$ ?

Answer 1

What ensures that $\lambda h \in [0,1]$ irrespective of the value of $\lambda$ ?

Nothing does, and this is not what these conditions say. Condition 1. should read $$ P(N_h=1)=\lambda h+o(h), $$ which means $$ \lim\limits_{h\to0+}\frac{P(N_h=1)}h=\lambda. $$ Note that $O(h)$ should read $o(h)$ and that the condition is only concerned with the limit when $h\to0+$.

Likewise, condition 2. should read $$ P(N_h\geqslant2)=o(h), $$ which means $$ \lim\limits_{h\to0+}\frac{P(N_h\geqslant2)}h=0. $$