What is the characteristic function used for?

Question

Im totally new to statistics , but what is the characteristic function for ? I do not get that. I was reading about the bell curve and the Central Limit Theorem , but I did not get what the characteristic function is suppose to be , where it comes from or what it is used for.

It seems to appear in the proof of the Central Limit Theorem, but I do not know why. It seems to have a Taylor series but again I do not know why.

I have been told all this relates to sum of 2 or 3 dices but again I do not understand how.

I know Bayes theorem (+proof) but I guess its unrelated ?

Sorry for the noob question. Thanks.

Answer 1

You can use it to show the sum of two independent normal random variables with means $\mu_1$ and $\mu_2$ and respective variances $\sigma_1^2$ and $\sigma_2^2$ is normal with mean $\mu_1+\mu_2$ and variance $\sigma_1^2+\sigma_2^2$. This is extremely easy using characteristic functions, less so if you try to do it directly.

Answer 2

Characteristic functions are great way to understand the sum of independent random variables. If $X_1,X_2\,\dots,X_n$ are independent random variables, and $Y = \sum_{k=1}^n X_k$, then $\chi_Y(s) = \prod_{k=1}^n \chi_{X_k}(s)$.

Note also that $\chi_{\alpha X}(s) = \chi_X(\alpha s)$.

Also its Taylor's series is $\chi_X(s) = \sum_{r=0}^\infty \frac{(is)^r E(X^r)}{r!}$.

This can be used to prove the central limit theorem. Here is the outline (with a lot of the rigour removed). Suppose $X_k$ is a sequence of iidrv with mean zero and variance $\sigma^2$. Let $Y_n = \frac1{\sqrt n}\sum_{k=1}^n X_k$. Then $$ \chi_Y(s) = [\chi_{X_1}(s/\sqrt n)]^n = \left[1-\frac{\sigma^2s^2}{2n} + O(n^{-3/2})\right]^n \to \exp(-\tfrac12 \sigma^2 s^2) $$ as $n \to \infty$. And $ \exp(-\tfrac12 \sigma^2 s^2)$ is the characteristic function of a $N(0,\sigma)$ random variable.