Characteristic Functions and motivations

I've recently studied characteristic functions in my probability course and I can't get why we define it to be the Fourier transform of the distribution (if the random variabile is continuous).

I mean that if $X$ is a random variable, $\varphi_X (t) = \mathbb{E}(e^{i t X}) = \int_{-\infty}^{+\infty} e^{i t x}f_X(x) dx$ where $f_X(x)$ is the distrubution function of $X$, and I can't see any motivation for doing this. I asked my professor but he wasn't clear at all; he said something like this:

"Since we proved the theorem that if $\varphi_X (t) = \varphi_Y (t)$ then $X \sim Y$ (or $P_X \equiv P_Y)$, it is natural to define it this way".

But of course, to prove that we need the definition! So I couldn't really make up my mind about it, if you could provide some help in this sense (motivation for defining the characteristic function of a random variable as the Fourier transform of its distribution) it would be much appreciated.

-

The reason we care about the Fourier transform of a distribution is that it has some useful properties. The term "characteristic function" is, of course, just a label.

Maybe a comparison with some of the other kinds of generating functions for random variables would be helpful. These include:

1. Probability generating function, $E[t^X]$ (also known as the factorial moment generating function),

2. Moment generating function, $E[e^{tX}]$ (which is the Laplace transform if $X$ is nonnegative),

3. Characteristic function, $E[e^{itX}]$ (which, as you have stated, is the Fourier transform).

To quote from Casella and Berger's Statistical Inference (first edition, p. 84),

"Perhaps the most useful of all these types of functions is the characteristic function... The characteristic function does much more than the mgf [moment generating function] does. When the moments of $F_X$ exist, $\phi_X$ can be used to generate them, much like an mgf. The characteristic function always exists [unlike the mgf] and it completely determines the distribution. That is, every cdf has a unique characteristic function."

They then give an example of a theorem that, for mgf's, has some qualifications to it, but that doesn't require any for characteristic functions.

To elaborate on the examples in the quote, the characteristic function has the two following useful properties (although it has more):

1. It can be used to find the moments $E[X^n]$,

2. It can be used to show that a transformation of a random variable from a particular distribution has some other known distribution. This method is often much easier than using the cdf technique for transformations. (See, for example, this answer to a recent math.SE question. The answer uses mgf's rather than characteristic functions, but the idea is similar.)

Practically speaking, the short answer is that it's convenient. The characteristic function has better analytic properties than the moment generating function, lets you study all of the moments of a random variable at once, and has the extremely convenient property that $\phi_{X+Y}(t) = \phi_X(t) \phi_Y(t)$ if $X, Y$ are independent. This makes the characteristic function an amazing tool for understanding sums of independent random variables, and indeed a standard proof of the central limit theorem proceeds via a computation of characteristic functions.