Self-similarity

• Paul Lévy generalized the Central Limit Theorem, by looking for distributions that were "stable", having 3 exclusive properties:

  • Invariant under addition: X1+X2+...+Xn= cnX + dn
  • Domain of attraction: cn= n1/Ñ ; 0<Ñ<2
  • Canonical characteristic function: H- or Fox-fcns.

• In other words, For a given Ñ, there exists a self-similar function, having a canonical form, that all random distributions will converge to.

• Lévy-stable probability distribution functions