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?

  • 3
    $\begingroup$ Yes it does, the characteristic function completely defines the probability distribution. $\endgroup$ – Ian May 15 '14 at 23:31

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.

| cite | improve this answer | |
  • $\begingroup$ Is there another way to prove it, without inversion ? $\endgroup$ – Olórin Oct 28 '19 at 16:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy