Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedaudited 2026-09-10
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Identical distribution without independence can defeat the mean law

Statement refuted

Identical integrable marginals alone do not imply the strong mean law: on {0,1} with equal masses, set X(ω)=ω and Xn=X for every n.

Counterexample

Given: The objects, hypotheses and definitions in the statement. Its conclusions are to be established below.

1.1

The law gives each atom mass 1/2 and has total mass one. Every coordinate has this same law, and EXn=0/2+1/2=1/2. It is integrable since |Xn|<=1.

givenalgebra
2.1

For every n, Sn/n=X at both points. Hence ∣Sn/n−EX1∣=1/2 everywhere, and convergence to the common mean fails on the whole space. Independence fails as well: P(X1=1,X2=1)=1/2≠(1/2)2.

givenalgebra∎

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