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Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$
If $\phi(t)$ is the characteristic function of a random variable $X$, then $\Re(\phi(t))$ is also a characteristic function of some random variable $Y$ (This can be easily seen by Polya's criterion). How can we find $Y$? ($\Re(\cdot)$ detones the real part of a complex number)
I have already read this related question but I cannot figure out the solution. How can we find the characteristic function of $AX+(1-A)(-X)$?
Let $A$ be independent of $X$ with $P(A=1)=P(A=0)=\frac{1}{2}$. Then $$ E\left[e^{it\{AX+(1-A)(-X)\}}\right]=\frac{1}{2}E\left[e^{itX}\right]+\frac{1}{2}E\left[e^{it(-X)}\right]=\frac{\phi(t)+\phi(-t)}{2}, $$ but using that $\cos$ is even and $\sin$ is odd, we obtain $$ \phi(-t)=E\left[e^{i(-t)X}\right]=E[\cos(-tX)]+iE[\sin(-tX)]=E[\cos(tX)]-iE[\sin(tX)] $$ and so $\phi(t)+\phi(-t)=2E[\cos(tX)]=2\Re(\phi(t))$, which yields the result.