kalman filter with time-varying noise? in regular discrete-time (1 dimensional) kalman filter, it is assumed that we have white gaussian noise affecting the transitions and the observations:

$x(t+1) = Ax + w$

$y(t) = Cx(t) + v$ 

assuming $w$ follows Gaussian(0, $\sigma_1$) and $v$ follows Gaussian(0, $\sigma_2$) distributions. 


will the standard kalman filter not work or not be optimal if the gaussian noise variables are time varying, meaning $w$ and $v$ become $w(t)$ and $v(t)$ and somehow change with time, becoming systematically bigger or smaller?
related: the kalman filter generally has the behavior that its variance is decreasing with time. will that be true even if $w(t)$ and $v(t)$ are are time-varying even if they are assumed not to be? will the kalman filter's variance still get smaller even if the data are sampled from time-varying noise process?
1) It depends on what you call the standard Kalman filter -- I will call the equations in the picture below to be the "standard Kalman filter". You can easily derive an expression for the Kalman filter where the covariance matrices of the noise processes are time varying in terms of the covariances of the noise processes (and cross-covariance of the noise processes). 

Moon and Stirling's Mathematical Methods and Algorithms for Signal Processing derives the Kalman filter under $w(t)$ has mean $0$, covariance $E[w(t) w(s)^T] = Q_t \delta_{ts}$, $v(t)$ is also mean zero with covariance $E[v(t) v(s)^T] = R_t \delta_{ts}$ and $E[w(t) v(s)^T] = M_t \delta_{ts}$ (if you assume $M(t) = 0$, the equations simplify considerably). 

For the system model 
$ x_{t+1} = A_t x_t + w_t , y_t = C_t x_t + v_t$  [$w_t = w(t), v_t=v(t)$] with $M_t = 0$, the following are the resultant Kalman equations (from Moon &amp; Stirling) to get the predicted state estimate (13.45) and filtered state estimate (13.47):



These equations can be shown to be optimal using either a Bayesian approach (MAP with Gaussianity assumptions) or Linear MMSE approach for the given system model. 

2) Convergence of the Kalman filter is a bit of an annoying subject. You can read these notes for more details or check the book Stochastic Systems: Estimation, Identification and Adaptive Control by Pravin Variya and P.R. Kumar or one of the other texts on stochastic systems. Essentially, if the systems are controllable and reachable and yada yada, you can get some statements on convergence properties.