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

Let
\{(X_i, Y_i,)\}_{i=1}^{\infty}
be a memoryless correlated source with finite alphabets, and let us imagine that one person, encoder 1, observes only
X^n = X_1,\cdots,X_n
and another person, encoder 2, observes only
Y^n = Y_1,\cdots,Y_n
. The encoders can produce encoding functions
f_n(X^n)
and
g_n(Y^n)
respectively, which are made available to the decoder. We determine the rate region in case the decoder is interested only in knowing
Y^n = Y_1,\cdots,Y_n
(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].