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Two new theorems show that small amounts of noise can increase the mutual information of threshold neurons that detect subthreshold signals. The first theorem shows that this "stochastic resonance" effect holds for all finite-variance noise probability density functions that obey a simple mean constraint that the user can control. The second theorem shows that this effect holds for all infinite-variance noise types in the broad class of stable distributions. Stable bell curves can model extremely impulsive noise environments. So the second theorem shows that this stochastic-resonance effect is robust against violent fluctuations in the additive noise process.