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Conditionally on past observations, adding the independent state increment gives the predictionThe new observation then has conditional distribution $N({\hat{S}}_{t-1},R_t+V)$, and its conditional covariance with $S_t$ is $R_t$. Equivalently, multiplying the state prediction probability density function by the observation likelihood and completing the square givesFor positive variances, rearrangement yields the scalar Gaussian Kalman recursionHere the innovation is the observation minus its predicted mean. The displayed covariance formula also handles zero variances whenever its denominator is nonzero; a completely deterministic prediction and observation require no probabilistic update. Independence of the new noises from the past and the initial state is the usual state-space assumption needed for this conditional calculation.

The state-space model as printed supplies only uncorrelated white noise. That is insufficient for the claimed normal conditional distributions. For example, take $S_0$ normal, independent process noises $w_t$ taking values $\pm \sqrt{W}$ equally often, and independent Gaussian observation noises. With $G=1$ and no previous observations, $S_1=S_0+w_1$ is a two-component shifted mixture distribution, not normal for $W>0$. Its characteristic function is $e^{i\mu s-P_0{s}^2/2}\cos (\sqrt{W}s)$, which has zeros, whereas a normal characteristic function has none. Similarly a non-Gaussian observation noise prevents the asserted conditional normal law for $X_t$.

The intended scalar Gaussian Kalman recursion requires $w_t\sim N(0,W)$ and $v_t\sim N(0,V)$ independent of each other and of the previous state and observation history. Assume these Gaussian independence conditions, $V>0$, $W\ge 0$, and the given normal filtering law. Conditional on the past, $G{S}_{t-1}+w_t$ is a sum of independent normals, henceThe new observation noise is independent of $S_t$ and the past, so $X_t\mid S_t,{\mathcal{F}}_{t-1}\sim N(F{S}_t,V)$. Write $S_t=m_t+{\eta }_t$ and $X_t=F{m}_t+F{\eta }_t+v_t$. The independent normal pair $({\eta }_t,v_t)$ shows directly thatThe covariance $F{R}_t$ comes from the common ${\eta }_t$ term. Applying the conditional multivariate normal distribution formula with $X_t$ as the conditioned coordinate gives the Kalman filter updateThe final expression verifies nonnegativity of the posterior variance.

For $F=G=1$, defineThe recursions become ${\hat{S}}_t=(1-{\alpha }_t){\hat{S}}_{t-1}+{\alpha }_t{X}_t$ and $P_t={\alpha }_{t}V$. If $P_t\to P$, then ${\alpha }_t\to \alpha =P/V$ and, with $c=W/V$,The nonnegative solution is the steady-state local-level Kalman gain:For $c=0$ the limit is zero; for $c>0$ it lies strictly between zero and one. These are exact conditional-law results under the additional Gaussian assumptions, not consequences of the weaker printed noise description alone.