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If $X,Y$ are identically distributed random variables, then I know that their characteristic functions $\phi_X$ and $\phi_Y$ are the same. Does the converse also hold?

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    $\begingroup$ Yes it does, the characteristic function completely defines the probability distribution. $\endgroup$
    – Ian
    May 15, 2014 at 23:31

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Yes, it is true as a consequence of the inversion formula $$\mu(a,b) +\frac 12\mu(\{a,b\}) = \frac 1{2\pi}\lim_{ t\to +\infty}\int_{-T}^T\frac{e^{ita} -e^{itb} }{it}\varphi_\mu (t)\mathrm dt,$$ valid for $a\lt b$.

If $\mu$ and $\nu$ have the same characteristic function, then $\mu([a,b])=\nu([a,b])$ for each $a\lt b$. This is true for finite disjoint unions of half-open intervals, and these sets characterize Borel probability measures on the real line.

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  • $\begingroup$ Is there another way to prove it, without inversion ? $\endgroup$
    – Olórin
    Oct 28, 2019 at 16:35

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