The performance of adaptive state estimators for linear dynamic systems is investigated. The adaptive state estimates are formed under the assumption that the unknown system parameter belongs to a finite set and is thus readily implementable. It is shown that, for the true parameter value in a prescribed region in the parameter space, the corresponding a posteriori probablity (or weighting coefficient in the adaptive estimator) converges exponentially in the vth mean (v > 1) and almost surely to unity. The analysis is based on the asymptotic per sample formula for the Kullback information function, which is derived in this paper. The significance of the analysis for applications is also examined.