On a limit distribution of characteristics in stationary regime for the linear assimilation problem

A commonly used linear assimilation problem when the model state vector is corrected by observed data through the system of linear equations is considered. This problem is formulated as a Markov chain problem. For this problem, convergence of transitional probability of the corresponding Markov chain is investigated and the sufficient conditions of this convergence are found out. A special case of series depending on a parameter when this parameter goes to zero is discussed and the limit theorem about convergence to Gaussian distribution for analysis of state vector characteristics for this case is proved. The mean value and variance of this distribution are determined. The paper discusses how these results can be applied to practical operational assimilation and forecasting.