  The main goal of the paper is to develop an estimate for the conditional
probability function of random stationary ergodic symbolic sequences with
elements belonging to a finite alphabet. We elaborate a decomposition procedure
for the conditional probability function of sequences considered as the
high-order Markov chains. We represent the conditional probability function as
the sum of multi-linear memory function monomials of different orders (from
zero up to the chain order). This allows us to construct artificial sequences
by method of successive iterations taking into account at each step of
iterations increasingly more high correlations among random elements. At weak
correlations, the memory functions are uniquely expressed in terms of the
high-order symbolic correlation functions. The proposed method fills up the gap
between two approaches: the likelihood estimation and the additive Markov
chains. The obtained results might be used for sequential approximation of
artificial neural networks training.
