Let $\{\xi_n \}_{n \geq 1}$ be i.i.d random variables taking values on $\mathbb{Z}$. Let $\xi_0 = 0$.
$S_n = \sum\limits_{i=1}^{n} \xi_i,$ where $S_0=0$
$Y_n = \sum\limits_{i=0}^{n} S_i$.
My question is whether $Y_n$ is a Markov chain. I know it has to satisfy $P(Y_{n+1} = y_{n+1}| Y_{n} = y_{n}, \ldots, y_0=y_{0}) = P(Y_{n+1} - y_{n+1}| Y_n = {y_n})$.
Intuitively, I know it is not a Markov chain since by knowing the full information of the previous states, I can predict the next state $Y_{n+1}$ better but I am having trouble formulating a mathematical proof.