Lukac’s Characteristic Functions

I was curious to see the proof for this theorem, so I’ve started reading Lukac’s Characteristic Functions. It’s a good book, but as written expresses absolutely no interest in the subject of probability. While about a subject that is interesting because of its connection with probability — a characteristic function is the Fourier transform of a probability distribution function — the discussion is entirely in terms of characteristic functions. For example, studying the characteristic functions for infinitely divisible distributions is motivated by the question of which characteristic functions have the property that every n-th root is also a characteristic function.