The characteristic function of a random variable is a powerful tool for analyzing the distribution of sums of independent random variables. In the same way in which the MGF was related to the Laplace transform the characteristic function is related to the Fourier transform of a function.
Since for every and it is clear that the expectation defined above exists and
This is easy to prove using the Jensen inequality (Lemma 4.14) for the convex function .
If X has a probability density then the characteristic ...