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Probability Transforms & Tail Behavior

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Abstractions about transforming and bounding probability distributions — generating-function-based encodings like the cumulant and Esscher transform, tail and concentration measures such as Bernstein inequalities, the Q-function, and tail dependence, and specific distributions like the Linnik distribution and probability weighting function.

7 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

  • Bernstein inequalities (probability theory) — Bernstein inequalities are exponential concentration bounds that control the deviation of sums of independent bounded random variables using both variance and a bound on individual magnitude.
  • Cumulant — Encode a probability law by the coefficients of the logarithm of its generating function, so independent sums become coefficientwise addition and joint cumulants isolate connected dependence.
  • Esscher transform — Exponentially tilt a probability law by a parameter and normalize by its moment-generating function, producing a new law that shifts event weights while preserving absolute-continuity structure on the finite domain.
  • Linnik distribution — A symmetric heavy-tailed probability distribution whose characteristic function has Linnik form.
  • Probability Weighting Function — Transform objective or stated probabilities into subjective decision weights, typically overweighting small probabilities, underweighting middle and high probabilities, and treating the zero and certainty boundaries nonlinearly.
  • Q-function — Map a real threshold to the upper-tail probability of a standard normal variable, equivalently one half of the complementary error function at the threshold divided by the square root of two.
  • Tail dependence — Measure whether two continuously distributed variables continue to co-exceed matched extreme quantiles by taking an upper- or lower-tail conditional-probability limit determined by their copula rather than their marginal scales.