I read the definition of Poisson process in Shereve's "Stochastic Analysis" which is constructed explicitly:
(A)
step 1:construct iid exponential r.v.$\tau_i$ with parameter $\lambda$
step 2:define $S_n=\sum_{k=1}^n \tau_k$.
step 3:define $N(t)=\max\{n:S_n\le t\}$
then $N(t)$ is called a Poisson process with parameter $\lambda$
But I wonder if there is a equivalent way to define the Poisson process according to its property
for example
(B)
$N(t)$ is a stochastic process,which satisfies
(1)counting process (nonnegative integer valued,increasing) with $N(0)=0$
(2)independent and stationary increment
(3)$\mathbb P(N(t)=n)=\frac{(\lambda t)^n}{n!}e^{-\lambda t}$
I wonder if the two definition (A) and (B) are equivalent.
$A\Rightarrow B$ is easy.
Now consider why $B\Rightarrow A$:
first I proved that each time $N(t)$ jump at most 1 each time according to B(3) and stationary increments.
then I can construct $S_k=$the time when $k$th jump happens
then define $\tau_k=S_k-S_{k-1}$
now I want to prove that $\tau_k$ is iid exponential distributed.
But I can only prove that $\tau_1$ is exponential distributed,how to do the next?