Noiseless coding of correlated information sources

Correlated information sequences <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\cdots ,X_{-1},X_0,X_1, \cdots</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\cdots,Y_{-1},Y_0,Y_1, \cdots</tex> are generated by repeated independent drawings of a pair of discrete random variables <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X, Y</tex> from a given bivariate distribution <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{XY} (x,y)</tex> . We determine the minimum number of bits per character <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_X</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_Y</tex> needed to encode these sequences so that they can be faithfully reproduced under a variety of assumptions regarding the encoders and decoders. The results, some of which are not at all obvious, are presented as an admissible rate region <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\mathcal{R}</tex> in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_X - R_Y</tex> plane. They generalize a similar and well-known result for a single information sequence, namely <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_X \geq H (X)</tex> for faithful reproduction.