Timeline for If $Z$ is standard normal and $f$ is analytic. Is $g(t)= E[ f(Z-t)]$ analytic?
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| Oct 9, 2017 at 15:42 | comment | added | Terry Tao | This argument seems fine to me (modulo a missing factor of $\frac{1}{\sqrt{2\pi}}$ and a sign error). The expectation ${\bf E} f(Z-t) = \frac{1}{\sqrt{2\pi}} \int f(x-t) e^{-x^2/2}\ dx$ can be rewritten as $\frac{1}{\sqrt{2\pi}} \int f(x) e^{-(x+t)^2/2}\ dx$ after a change of variables $x \mapsto x+t$. One now has complex analyticity from Morera's theorem; one could also proceed in a real variable fashion by Taylor expansion and Fubini. | |
| Aug 10, 2017 at 14:32 | comment | added | Boby | You can not choose you $f$ to be like this. First, you have to specify $f(x)$ and then you plug it into the expectation by setting $x=Z-t$. You can not do this with the function you picked. | |
| Aug 10, 2017 at 14:04 | history | undeleted | user83457 | ||
| Aug 10, 2017 at 14:02 | history | deleted | user83457 | via Vote | |
| Aug 10, 2017 at 14:01 | history | answered | user83457 | CC BY-SA 3.0 |