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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)

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