For Steinhaus random multiplicative functions, the sum over A converges to CN(0,1) only if |A|=o(N), with a sharp version for most sets of density rho where (1-rho)^{-1}=o((log log N)^{1/2}) using an extra sqrt(1-rho) factor.
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Escaping Chaos in Random Multiplicative Functions
For Steinhaus random multiplicative functions, the sum over A converges to CN(0,1) only if |A|=o(N), with a sharp version for most sets of density rho where (1-rho)^{-1}=o((log log N)^{1/2}) using an extra sqrt(1-rho) factor.