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In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

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Gaussian approximation of the characteristic function of Rademacher sum

A general comment which admittedly doesn't answer the question: One cannot choose the $a_i$ so that the sum would be Gaussian. For one can break up the sum into two independent pieces, and necessarily …
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