The stability problem of Kalman filtering for linear stochastic systems with scheduled measurements in  is reconsidered in this paper. The transmission of a vector observation from the sensor to the remote estimator is realized by sequentially transmitting each component of the observation to the estimator in one time step. The communication of each component is triggered if and only if the corresponding component of normalized measurement innovation vector is larger than a given threshold. As a complementary to , we extend the measurement data scheduler to have different thresholds assigned to different components of the normalized measurement innovation vector and similarly derive the sequential Kalman filter. Moreover, the sufficient and necessary conditions for guaranteeing the stability of mean squared estimation error are established for general linear systems by explicitly investigating the convergence properties of a specially constructed axillary function.