I want to find the Covariance function of a stochastic Process $(X(t))_{t \in I}$ for $X(t) := \sum_{l=1}^k\frac{Y(t-l)}{2^l}$ for some $k\in\mathbb{N}$ and $(Y(t))_{t \in \mathbb{Z}}$ uncorrelated and identically distributed with $\mathbb{E}Y(0)= 0$ $\mathbb{E}Y(0)^2 = \sigma^2>0$ and $I = \mathbb{Z}$.
The covariance function is given by
$$Cov(X(s),X(t)) = \mathbb{E}[(X(s)-\mathbb{E}X(s)) \cdot (X(t)-\mathbb{E}X(t))] = \mathbb{E}[(X(s)X(t))] - \mathbb{E}[X(s)]\mathbb{E}[X(t)]$$
I can't even really figure out how to find $\mathbb{E}[X(t)]$ when I know so little about $Y$ and I wouldn't even know where to start to find $\mathbb{E}[(X(s)X(t))]$. Can anyone help?
$Y(t)$ is identically distributed and $EY(0)=0$ so $EY(t)=0$ for all $t$. This implies $EX(t)=\sum \frac {EY(t-l)} {2^{l}}=0$. Now $EX(s)X(t)=E[\sum\limits_{l=1}^{k}\frac {Y(t-l)} {2^{l}}][\sum\limits_{j=1}^{k}\frac {Y(s-j)} {2^{j}}]$. Just multiply this out and use the fact that $EY(t-l)Y(s-j)=0$ if $t-l \neq s-j$ and $EY(t-l)Y(s-j)=\sigma^{2}$ if $t-l = s-j$ .