Main Question :

In an article I was reading there is a recursive relation suggested to obtain the moment generating function of a random variable by setting: \begin{align*} M_1(s) &= \frac{a_1}{s-a_1} \frac{M_X(a_1)-M_X(s)}{1-M_X(a_1)},\\ M_n(s) &= \frac{a_n}{s-a_n} \left( M_X(a_n) \frac{1-M_{n-1}(s)}{1-M_{n-1}(a_n)} - M_X(s) \right), \end{align*} the values that are used here are explained below. My main question is now: $M_X$ and $M_1, M_2\dots$ are moment generating functions, but as we know a moment generating function need not exist. Is there a standard (or non-standard) method to obtain a recursive relation for some defining function for the random variables which is guaranteed to exist from the recursive relation? Like for the Characteristic function?

Some more details

here $a_1 > a_2> \dots$ are some fixed numbers that satisfy $\sum_n a_n <1$ (take for example $a_1=1/2, a_2=0,\dots$ and $X$ is some random variable with MGF $M_X$. Using this scheme one finds successively the moment generating functions $M_n$ of some random variables. The recursion ends after the first time $a_n=0$, i.e. if $a_n>0$ and $a_{n+1} \neq 0$, then $M_n$ is the last moment generating function we compute.


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