The ecf is $\phi_n(\omega) = \frac{1}{n}\sum_{j=1}^ne^{iX_j\omega}$. I'm stuck on trying to see why the following is true $$|\phi_n(\omega)|^2 = \phi_n(\omega)\phi_n(-\omega)$$ Wouldn't this imply that $\phi_n(\omega)=\phi_n(-\omega)$?
2026-03-27 01:14:00.1774574040
Empirical characteristic function
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It's true because $\phi_n(-\omega)=\overline{\phi_n(\omega)}$ (as long as $\omega$ is real not complex). Thus, you get $$ |\phi_n(\omega)|^2 =\phi_n(\omega)\overline{\phi_n(\omega)} =\phi_n(\omega)\phi_n(-\omega). $$