I'm trying to understand the usage of this method.
What i mean: (maybe i named this method in wrong way)
Let $\xi$ be a random variable with distribution function $F(x)$
Add condition: $\mathbb{E}e^{t\xi}\lt \infty$
Denote $S(x) = \int\limits_{-\infty}^x e^{ts}dF(s)$
After all, let $G(x) = \frac{S(x)}{\mathbb{E}e^{t\xi}}$
If $\eta$ is a random variable with distribution function $G(x)$, we can say, that $\eta$ and $\xi$ have conjugate distributions.
Can anybody help me with this interesting method? (usage and features) (Or good books about it)