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A constant posterior variance must be a fixed point of the recursion in part (b). Multiplying $P=V(P+W)/(P+W+V)$ by its positive denominator givesConversely, if a nonnegative $P$ solves this equation and the filter is initialized with $P_0=P$, the recursion preserves $P_t=P$ at every step. The fixed-point equation alone does not force an arbitrary initial covariance to be constant.

For $W,V>0$, the admissible root and its steady-state local-level Kalman gain areThe state estimate then obeys exponential smoothing:Iteration makes the weights explicit:Thus older observations receive geometrically decreasing weights. Consistently with part (a), the innovations variance is $q=P+W+V$ and the differenced MA coefficient is $\vartheta =-(1-K)$. The degeneracies $W=0,P=0$ and $V=0,P=0$ give gain zero and gain one respectively when the denominator is positive.