Lévy's continuity theorem implies that if the sequence of characteristic functions $(\varphi_n)_n$ of a sequence of random variables $(X_n)_n$ converges pointwise to the characteristic function $\varphi$ of some random variable $X$, then $X_n$ converges in distribution to $X$.

Is there a quantitative version of this statement in the form of an upper bound on the the pointwise distance between the corresponding distribution functions $F_n$ and $F$ in terms of some distance between $\varphi_n$ and $\varphi$?

Continuity of $F$ can be assumed.


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