Multiplication Property of Characteristic Functions

Theorem

Let $X$ and $Y$ be independent random variables on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $\phi_X$ and $\phi_Y$ denote the characteristic functions of $X$ and $Y$ respectively.

Then:

$\phi_{X + Y} = \phi_X \phi_Y$

Let $i = \sqrt{-1}$.

Let $\expect X$ denote the expectation of $X$.

$\ds \map {\phi_{X + Y} } t$

$=$

$\ds \expect {e^{i t \paren {X + Y} } }$

$\ds $

$=$

$\ds \expect {e^{i t X} e^{i t Y} }$

$\ds $

$=$

$\ds \expect {e^{i t X} } \expect {e^{ i t Y} }$

$\ds $

$=$

$\ds \map {\phi_X} t \map {\phi_Y} t$

Hence:

$\phi_{X + Y} = \phi_X \phi_Y$

$\blacksquare$