Dispersion Bound for the Wyner-Ahlswede-Körner Network via Reverse Hypercontractivity on Types
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Using the functional-entropic duality and the reverse hypercontractivity of the transposition semigroup, we lower bound the error probability for each joint type in the Wyner-Ahlswede-Körner problem. Then by averaging the error probability over types, we lower bound the c-dispersion (which characterizes the second-order behavior of the weighted sum of the rates of the two compressors when a nonvanishing error probability is small) as the variance of the gradient of _P_U|X{cH(Y|U)+I(U;X)} with respect to Q_XY, the per-letter side information and source distribution. On the other hand, using the method of types we derive a new upper bound on the c-dispersion, which improves the existing upper bounds but has a gap to the aforementioned lower bound.
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