Lévy continuity theorem
= Lévy continuity theorem
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If the <characteristic functions> $\varphi_n$ of <probability distributions> $\mu_n$ converge pointwise to a function $\varphi$ that is continuous at zero, then $\varphi$ is a characteristic function and $\mu_n$ <convergence in distribution>[converges in distribution] to its distribution. Conversely, convergence in distribution implies pointwise convergence of the characteristic functions.