I am facing the following problem.
Let $X,Y$ be independent random variables with standard normal distribution. Find the characteristic function of a variable $ XY $.
I have found some information, especially the fact that if $ \phi_X,\phi_Y $ denote characteristic functions, then
$$ \phi_{XY}(t) = \mathbb{E}\phi_X(tY).$$
The only problem is that the proof required knowledge of conditional expectation which I do not have. Is there and around way?
Best Answer
By independence, $P_{(X,Y)}=P_X\otimes P_Y$, and hence $$ \phi_{XY}(t)=\int_{\mathbb{R}^2}\mathrm{e}^{itxy}\,P_{(X,Y)}(\mathrm dx,\mathrm dy)=\int_{\mathbb{R}^2}\mathrm{e}^{itxy}P_X\otimes P_Y(\mathrm dx,\mathrm dy). $$ The complex version of Fubini's theorem now allows us to write this as an iterated integral $$ \begin{align} \phi_{XY}(t)&=\int_\mathbb{R} \Big(\int_\mathbb{R} \mathrm{e}^{itxy} \,P_X(\mathrm dx)\Big)P_Y(\mathrm dy)=\int_\mathbb{R} \mathrm{E}[\mathrm{e}^{ityX}]\,P_Y(\mathrm dy)\\ &=\int_\mathbb{R}\phi_X(ty)\,P_Y(\mathrm dy)=\mathrm{E}[\phi_X(tY)]. \end{align} $$