# Proving that a process is a Poisson Point Process

I'm stuck proving the following proposition:

Let $\{E_i\}_{i\geq 1}$ be i.i.d. exponential random variables on $[0,\infty)$ with parameter $1$: $P(E_i > x)= e^{-x}, x>0.$ Let $\Gamma_{n} =\sum_{i=1}^{n} E_i,n\geq 1.$ Show that $$N=\sum_{n\geq 1}\delta_{\Gamma_{n}},$$ is a homogeneous Poisson process on $[0,\infty).$

Because of the hypotesis, $\Gamma_{n}$ has distribution $\Gamma(n,1).$ Next, I tried to compute the "rate" or mean of the process, but I found with a hard series to calculate.

Any kind of help is thanked in advance.

• The notation seems difficult. I think you mean this is $N(t)$, being the number of arrivals up to time $t$. BUt, this is the definition of a Poisson process (i.e., a counting process having iid exponential inter-arrival times). Are you working with some alternative definition? What is it you are trying to prove, exactly? Some other definitions might involve "independent increments" and so on... – Michael May 22 '17 at 0:18
• @Michael The notation means that $N$ is the random measure on $S=[0,+\infty)$ such that, for every Borel subset of $S$, $N(B)$ is the size of the (random) set $\{n\geqslant1\mid\Gamma_n\in B\}$. To recover the (integer valued) random variables $N(t)$ you might be more accustomed to, simply consider $N([0,t])$. (And I fully concur with your remark about the necessity for the OP to expand the definitions they are using.) – Did May 22 '17 at 13:32