Source coding with side information and a converse for degraded broadcast channels

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{(X_i, Y_i,)\}ᵢ₌₁∞</tex> be a memoryless correlated source with finite alphabets, and let us imagine that one person, encoder 1, observes only <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X^n = X_1,⋯,X_n</tex> and another person, encoder 2, observes only <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y^n = Y_1,⋯,Y_n</tex> . The encoders can produce encoding functions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f_n(X^n)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g_n(Y^n)</tex> respectively, which are made available to the decoder. We determine the rate region in case the decoder is interested only in knowing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y^n = Y_1,⋯,Y_n</tex> (with small error probability). In Section H of the paper we give a characterization of the capacity region for degraded broadcast channels (DBC's), which was conjectured by Bergmans [11] and is somewhat sharper than the one obtained by Gallager [12].