While I am editing the paper, some of the reviews mentioned an important paper, Kalman Filtering With Intermittent Observations. The paper mainly deals with the stochastic nature of observation availability. They derived an expected covariance update, named modified Riccati recursion.

However, I think that there is an error while taking the uncertainty to the infinity. If we apply matrix inversion lemma first, the resulting covariance update is still an ordinary Riccati recursion. I think it is easy to verify which one is correct, even in the simpliest scalar case.

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