On approximating information sources

In this paper we introduce a class of information sources T = [Si A], i = 1,2, .....m, such that each source in this class approximates, in a defined sense, a given source S. For this purpose we use the set of product approximations for a distribution introduced by Lewis (Ref. 2.). Thus, for any source Si A ¿ T, the nth order distribution is a product of some lower order distributions corresponding to the source S and further the product is an extension of the lower order distributions. It is then shown that each source Si A ¿ T is "compatible" with the source S, where the nation of "compatibility" roughly means that each source Si A ¿ T can use a code designed for the source S equally effectively as regards the probability of incorrectly receiving a given sequence and still maintain an effective information rate which can be made arbitrarily close to the rate corresponding to the source Si A by increasing the length of the sequences. A precise definition of "compatibility" and the corresponding necessary and sufficient conditions have been stated previously (Ref. 3.).