I am trying to understand the link between the moment-generating function and characteristic function. The moment-generating function is defined as:
$$
M_X(t) = E(\exp(tX)) = 1 + \frac{t E(X)}{1} + \frac{t^2 E(X^2)}{2!} + \dots + \frac{t^n E(X^n)}{n!}
$$
Using the series expansion of $\exp(tX) = \sum_0^{\infty} \frac{(t)^n \cdot X^n}{n!}$, I can find all the moments of the distribution for the random variable X.
The characteristic function is defined as:
$$
\varphi_X(t) = E(\exp(itX)) = 1 + \frac{it E(X)}{1} – \frac{t^2 E(X^2)}{2!} + \ldots + \frac{(it)^n E(X^n)}{n!}
$$
I don't fully understand what information the imaginary number $i$ gives me more. I see that $i^2 = -1$ and thus we don't have only $+$ in the characteristic function, but why do we need to subtract moments in the characteristic function? What's the mathematical idea?
Best Answer
As mentioned in the comments, characteristic functions always exist, because they require integration of a function of modulus $1$. However, the moment generating function doesn't need to exist because in particular it requires the existence of moments of any order.
When we know that $E[e^{tX}]$ is integrable for all $t$, we can define $g(z):=E[e^{zX}]$ for each complex number $z$. Then we notice that $M_X(t)=g(t)$ and $\varphi_X(t)=g(it)$.