Paul Lévy continuity theorem
Let be distribution functions with characteristic functions , respectively. If converges pointwise to a limit , and if is continuous at , then there exists a distribution function such that weakly (http://planetmath.org/ConvergenceInDistribution), and the characteristic function associated to is .
Remark. The reciprocal of this theorem is a corollary to the Helly-Bray theorem; hence weakly if and only if pointwise; but this theorem says something stronger than the sufficiency of that : it says that the limit of a sequence of characteristic functions is a characteristic function whenever it is continuous at 0.
|Title||Paul Lévy continuity theorem|
|Date of creation||2013-03-22 13:14:31|
|Last modified on||2013-03-22 13:14:31|
|Last modified by||Koro (127)|