Sequences achieving the boundary of the entropy region for a two-source are virtually memoryless (Corresp.)

For a joint distribution{rm dist}(X,Y), the functionT(t)=min { H(Y|U): I(U wedge Y|X)=O, H(X|U)geq t}is an important characteristic. It equals the asymptotic minimum of(1/n)H(Y^{n})for random pairs of sequences(X^{n}, Y^{n}), wherefrac{1}{n} sum ^{n}_{i=1}{rm dist} X_{i} sim {rm dist} X, {rm dist} Y^{n}|X^{n} = ({rm dist} Y|X)^{n}, frac{1}{n}H(X^{n})geq t.We show that if, for(X^{n}, Y^{n})as given, the rate pair(1/n)H(X^{n}),(1/n)H(Y^{n})]approaches the nonlinear part of the curve(t,T(t)), then the sequenceX^{n}is virtually memoryless. Using this, we determine some extremal sections of the rate region of entropy characterization problems and find a nontrivial invariant for weak asymptotic isomorphy of discrete memoryless correlated sources.