Majority is Stablest for Dependent Variables
summary
Extends Majority is Stablest to dependent X,Y by proving for rho-correlated joint laws that E[J(f(X),g(Y))] ≤ J(Ef,Eg) plus an explicit error controlled by the sum of martingale-difference cubed terms; proof uses a two-point Taylor expansion, a hessian condition called rho-concavity and tensorization/induction, then hypercontractivity to convert the error into bounds on influences. Applications include parallel evolutions of Markov chains and high-temperature Glauber dynamics; requirement: existence of a filtration with stepwise rho bounds, error is third-order so it vanishes under CLT-type smoothing.
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