Verify the markov property for a sum of processes

Question

I'm given such a problem and I'm seeking for help.
Let {$X_n:n>1$} be an i.i.d. process with Poisson marginal PMF $p_x(k)$ = $e^{-\lambda}\frac{\lambda^k}{k!}$, $k \geq 0$, and let $N_l = \sum_{k=1}^{l}X_k$ and where $N_0 = 0$. Is $\{N_l\}_{l\geq 0}$ a Markov process? Explain in detail.
I'm wondering what the random process with poisson marginal pmf means, does it mean that X is a poisson process? Then if this is true, then I can try to prove the sum of poisson processes is also poisson, then N is a homogeneous poisson process, next homogeneous poisson process is markov process. I'm not sure whether this is the right way.
Can someone give me some help on this one? Thanks.

Answer 1

You can use the following fact:

If $(X_n)_n$ are independent random variables and the stochastic process $(Y_n)_n$ respects an identity of the form $Y_{n+1} = f_n(Y_n,X_{n+1})$, then $(Y_n)_n$ is a markov process.

Proof:

$\mathbb{E}[\phi(Y_{n+1})|Y_1,\ldots,Y_n] = \mathbb{E}[\phi(f_n(Y_n,X_{n+1}))|Y_1,\ldots,Y_n] = \mathbb{E}[\phi(f_n(y,X_{n+1}))]_{y=Y_{n}}$, which is $\sigma(Y_n)$-measurable.

Can you finish now?