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Top-down: divisibility is the operation (summation is an inverse) > arithmetic and number theory including > the statistical properties of numbers (how they converge and convolve). Wintner etc.
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In reverse: the distributions can be 'stable' if they are summable (divisible) > classes of distributions behaving like numbers > their 'arithmetic'. Zolotarev, Linnik, Khinchine.
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Bernstein Convolution as an operation of composition replacing summation for dependent variables. Schwartz, 'vector measures'. 3.5 Independence, Kac and Viet formula.
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A vision of unities splitting into (p+q) and adding as links to a chain of (p+q) and the error function integral... Weierstrass, Bernstein.