Steady-state local-level Kalman gain (${\alpha }_{\infty }=(\sqrt{c^2+4c}-c)/2$)

For the state-space model with both scalar coefficients equal to one, ${\alpha }_t=(P_{t-1}+W)/(P_{t-1}+W+V)$ and $P_t={\alpha }_{t}V$. If the variance converges and $c=W/V$, its limiting gain satisfies ${\alpha }^2+c\alpha -c=0$. The nonnegative root lies below one for finite $c>0$; it tends to zero with $c$ and to one as $c$ grows. Thus noisier state evolution assigns more weight to the latest observation.