Characteristic Functions — Section 9: Advanced Probability Tools

The characteristic function of $X$ is the Fourier-flavored cousin of the MGF:

\varphi_X(t) = E[e^{itX}]

Unlike the MGF, the characteristic function exists for every random variable, because $|e^{itX}| = 1$ everywhere.

Three properties make it the more general tool:

Always exists, even for heavy-tailed distributions like Cauchy where the MGF blows up.
Convergence in distribution is equivalent to pointwise convergence of characteristic functions (Lévy's theorem).
Lévy's inversion formula recovers the CDF from $\varphi$ .

A nice illustration: the Cauchy distribution has no finite moments, so its MGF doesn't exist. But its characteristic function is $\varphi(t) = e^{-|t|}$ . Sums of independent Cauchys multiply characteristic functions and remain Cauchy — explaining the strange property that averages of independent Cauchy variables don't shrink the spread.

The characteristic function of $X$ is the Fourier-flavored cousin of the MGF:

\varphi_X(t) = E[e^{itX}]

Unlike the MGF, the characteristic function exists for every random variable, because $|e^{itX}| = 1$ everywhere.

Three properties make it the more general tool:

Always exists, even for heavy-tailed distributions like Cauchy where the MGF blows up.
Convergence in distribution is equivalent to pointwise convergence of characteristic functions (Lévy's theorem).
Lévy's inversion formula recovers the CDF from $\varphi$ .