Given a random variable $X$, consider the k-th (non-central) moment about $a$, $E \left[ ( X – a )^k \right]$
Is there any relation of this value to the chracteristic function of variable's probability distribution like there is for the raw moments? i.e. do we have any identity similar to this one? $$E[X^k] = (-i)^k \varphi_X^{(k)}(0)$$ where $\varphi_X(\cdot)$ is the characteristic function of the random variable $X$
I mean if I know the characteristic function, calculating the non central moments should be just a simple application of the binomial theorem(or am I missing something?), but what I am really after is an understanding about what concrete effects the rest of the characteristic function has on the distribution. This identity makes it almost seem like the only important point it at zero.
Best Answer
Recall that $\varphi_X(t) = E[e^{itX}]$. Hence, $\varphi_{X-a}(t) = E[e^{it(X-a)}] = e^{-ita}\varphi_X(t)$. You can compute the moments using your formula for $\varphi_{X-a}(t)$.
Regarding your second question, I guess it corresponds to: "When is a distribution completely specified by it's moments?". An answer is here.
PS: Any analytic function is completely defined by it's derivatives on one point! So, if the characteristic function is analytic, the derivatives on $0$ have all the information about the characteristic function and the characteristic function has all the information about the distribution!