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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!

Community
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slaaidenn
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1 Answers1

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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).

angryavian
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  • Just a slight supplement: $\phi_{X_k}(t)=\mathbb E[e^{itX_k}]=\int_{\mathbb R} e^{itx}d\mu_{X_k}(x)$, where $d\mu_{X_k}$ is the distribution (measure) induced by $X_k$. Now it is more clear to see the convergence of characteristic function follows from the Portmanteau lemma. – Sam Wong Apr 18 '23 at 10:20