Assume we specify a state space model as

$$Y_t = a X_t + W_t$$

and

$$X_{t+1} = b X_t + V_t$$

where $b,a \in R$, $E[W_t] = E[V_t] = 0 \quad \forall{t }$ and $W_t $ and $V_t$ are indipendent for all t and both have finite second moment.

Notice that $W_t$ and $V_t$ are not assumed to be Gaussian.

I think I once saw somewhere a derivation of a filter à la Kalman filter with these assumptions only, has this been tried? Does somebody have any references?

EDIT: just to be clear, I am looking for a reference to the derivation of a filter with only these hypothesis, I have seen derivations of the Kalman filter but the ones I have seen all rely on the Gaussian assumption.