moments and derivatives of the characteristic function

The characteristic function is a moment-generating device that, unlike the moment generating function E[e^(tX)], always exists. The precise link between moments of X and derivatives of phi at the origin is the calculus that powers cf proofs of the CLT and of refined corrections; the question is exactly how much moment information the cf encodes near t=0.

If E[|X|^k] < infinity, then phi(t) = E[e^(itX)] is k-times continuously differentiable and you may differentiate under the expectation: phi^(j)(t) = E[(iX)^j e^(itX)], so at the origin phi^(j)(0) = i^j E[X^j] for j = 0, 1, ..., k. This gives the Taylor expansion at 0, phi(t) = sum_{j=0}^{k} (it)^j E[X^j]/j! + o(t^k), the engine behind every cf-based limit theorem: for mean-0 variance-sigma^2 variables, phi(t) = 1 - sigma^2 t^2/2 + o(t^2), which raised to the n-th power after scaling gives the Gaussian limit. The reverse direction is subtler and order-dependent: a finite even derivative phi^(2m)(0) does imply E[X^(2m)] < infinity, but the existence of an ODD derivative phi^(2m+1)(0) does NOT force E[|X|^(2m+1)] < infinity.

This correspondence matters because it turns moment hypotheses into clean polynomial expansions of phi near 0 that you simply substitute and take powers of. The asymmetry between even and odd orders is the honest subtlety: it is why CLT statements are usually phrased with even moments (variance) and why the Berry-Esseen rate needs the third ABSOLUTE moment E[|X|^3], not the third moment itself. Also note the cf encodes moments only locally at 0; global smoothness of phi away from 0 reflects the density's smoothness, a separate (smoothness-tails) matter.