Let $X_n$ be a independent identically distributed sequence of integer valued random variables. Suppose $S_n = \sum_{k=1}^n X_k$ with $S_0=0$, and $Z_n = \sum_{j=1}^n S_j$.
Does $(Z_n)$ form a Markov chain, why or why not: I get the following: Since $(S_n)$ is a Markov chain,
$$ \begin{eqnarray*} P(Z_{n+1}=z_{n+1} |Z_n=z_n,\dots, Z_1=z_1) &=&P(S_{n+1}=z_{n+1}-z_n |S_n=z_n-z_{n-1},\dots, S_1=z_1)\\ &=& P(S_{n+1}=z_{n+1}-z_n|S_n=z_n-z_{n-1})\\ \end{eqnarray*} $$ So it seems like the Markov condition shouldn't hold for $Z_n$, but I'm having difficulties coming up with a counter example.