why the characteristic function always exists
It is a striking claim: the moment generating function can fail to exist, but the characteristic function never does — for every random variable on earth, no matter how wild its tails, phi_X(t) is defined and finite. Why does swapping the real exponent for an imaginary one, e^(tX) becoming e^(itX), make all the difference? The answer is a one-line observation about size.
Compare the two integrands. For the mgf, e^(tX) grows exponentially as X grows; if the distribution has enough probability far out in the tail, the average E[e^(tX)] is an average of arbitrarily huge numbers and diverges to infinity. For the characteristic function, e^(itX) is completely different. By Euler's formula e^(itX) = cos(tX) + i sin(tX), and a quick check gives |e^(itX)|^2 = cos^2(tX) + sin^2(tX) = 1. So |e^(itX)| = 1 for every value of X and t — the integrand is a unit-length complex number that merely rotates, never grows. Averaging quantities that all have size 1 can only produce something of size at most 1; the average cannot run off to infinity. Hence E[e^(itX)] exists and |phi_X(t)| <= 1 always.
This is exactly why the characteristic function is the rigorous foundation of probability theory rather than a mere convenience. Heavy-tailed laws like the Cauchy, which have no mean, no variance, and no mgf, still have a perfectly good characteristic function. So any theorem you prove with characteristic functions applies universally, with no 'provided the mgf exists' clause attached. The bounded modulus is the whole reason transforms can carry the deepest limit theorems, including the most general forms of the central limit theorem.
The Cauchy distribution has tails so heavy that E[X] does not exist and E[e^(tX)] diverges for every t != 0, so it has no usable mgf. Yet its characteristic function is the perfectly tame phi_X(t) = e^(-|t|), bounded by 1 — vivid proof that |e^(itX)| = 1 saves the day where exponential growth dooms the mgf.
Because |e^(itX)| = 1, the average E[e^(itX)] is bounded by 1 and always exists — even when the mgf does not.
Existence does not mean every moment exists. phi_X(t) is always defined, but it is only differentiable enough at 0 to read off the moments that the distribution actually has — the Cauchy's phi has a corner at 0, matching its missing mean.