Levy's Continuity Theorem

Levy's Continuity Theorem relates the convergence in distribution of a sequence of random variables to the pointwise convergence of their characteristic functions, stating that the two notions of convergence coincide whenever the pointwise limit of the characteristic functions is itself continuous at the origin. Named for Paul Levy, it is a foundational tool of probability theory used to prove convergence results, including the central limit theorem, by working with characteristic functions rather than distributions directly.