Let $(\mathbb{Z}^+, \mathcal{P}(\mathbb{Z}^+))$ be a measurable space. Let $(\Omega,\mathcal{F},P )$ be a probability space. Let $\{Y_{n,k}: (\Omega, F) \rightarrow (\mathbb{Z}^+, \mathcal{P}(\mathbb{Z}^+)): n \in \mathbb{Z}^+, k \in \mathbb{N} \}$ be a collection of iid copies of $\mathbb{Z}^+$ valued random variables. For $i \in \mathbb{Z}^+, A \in \mathcal{P}(\mathbb{Z}^+))$, define $p : \mathbb{Z}^+ \times \mathcal{P}(\mathbb{Z}^+) \rightarrow [0,1] $ by $p(i,A)= P(\sum_{k=1}^i Y_{1,k} \in A)$.
Fix $x_0 \in \mathbb{Z}^+$ and define random variables $X_0=x_0$ and $X_{n+1}= P(\sum_{k=1}^{X_n} Y_{n,k} $ for $ n \in \mathbb{Z}^+ $.
Let $\mathcal{F}_n = \sigma\{ Y_{k,j} : k<n, j \in \mathbb{N}\}$. Show that $\{ X_n: n \in \mathbb{Z}^+\}$ is a Markov Chain with respect to $(\mathcal{F}_n)$ with transition probability $p$ and initial distribution $\delta_{x_0}$.
What I have done: I showed that $p$ is a transition probability on $(\mathbb{Z}^+, \mathcal{P}(\mathbb{Z}^+))$. And $X_n$ is $F_n$ adapted. From here, I'm not sure how to proceed and conclude.
Please help. Thank you!