Levy's continuity theorem says that if a sequence of random variables $X_{i}$ has characteristic functions $\phi_{X_i}$ converging pointwise to a characteristic function $\phi_{X}$ of other random variable $X$, then $X_i$ converges in distribution to $X$, i.e. the CDF's $F_{X_i}$ converges pointwise to $F_{X}$ in every continuity point of $F_{X}$.
Does the converse hold? If $F_{X_i}$ converges pointwise to $F_{X}$ in every continuity point of $F_{X}$, then $\phi_{X_i}$ converges pointwise to $\phi_{X}$?
This answer uses this result? Thanks!
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$\begingroup$Yes, the converse is the "easy" direction (if you have the Portmanteau lemma).
If $X_i \overset{d}{\to} X$ then by the Portmanteau lemma, $\phi_{X_i} \to \phi_X$ (since $x \mapsto e^{itx}$ is a bounded continuous function).
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