Let $X_k$ for $k=1,2,...$ be a sequence of i.i.d. random variables with $\mu_k=0$ and $\sigma_k^2=1$ for all $k$. Consider de random process $$S(t)=\sum\limits_{k=1}^{N(t)}X_k $$ where $N(t)$ is a Poisson process of mean $\lambda t$ and independent from $X_k\forall k$.
With this information, I have to calculate the autocorrelation
$$R(S(t_1),S(t_2))=\mathbb{E}[S(t_1)S(t_2)],t_2>t_1$$
So far, I've done this:
$$\mathbb{E}[S(t_1)S(t_2)]=\mathbb{E}[S(t_1)\cdot (S(t_2)+S(t_1)-S(t_1))]=\mathbb{E}[S^2(t_1)]+\mathbb{E}[S(t_1)(S(t_2)-S(t_1))]$$
The doubt I have is if the last term can be divided into the product of two expectations. I'm not sure if $S(t_1)$ and $S(t_2)-S(t_1)$ are independent from each other. Are they? If not, how can I solve this?