Is there any sequence of probability distributions such that their characteristic functions converge pointwise, but the sequence of prob. distributions itself does not converge weakly? :S
2026-03-30 13:34:06.1774877646
Characteristic function converges pointwise
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Yes, for example when $\mu_n$ is normally distributed with mean $0$ and standard deviation $n^2$: the sequence of characteristic functions converges pointwise to the map $\varphi$ such that $\varphi(t)=0$ if $t\neq 0$ and $\varphi(0)=1$. But such a sequence of measure cannot converge weakly (it's not tight).
However, if the sequence of characteristic functions converges pointwise to a function which is continuous at $1$, then one of the numerous theorems of Lévy gives weak convergence.