We call $(X_t)_{t\in\mathbb{Z}}$ a white noise series if $E[X_t] = 0$, $E[X_t^2] = \sigma^2$ and $E[X_{t+h}X_t] = 0$ for every $t \in \mathbb{Z}$ and every $h \in \mathbb{Z}\setminus\{0\}$.
I consider the natural filtration generated by a white noise series $X$, $\mathcal{F}_t = \sigma(X_s, s\leq t)$. I would like to know what $X_t$ should look like so that $$E[X_{t+k}X_{t+l}\mid \mathcal{F}_t] \neq 0$$ for some $k > l > 0$.