The Depoissonisation quintet: Rice-Poisson-Mellin-Newton-Laplace

This paper is devoted to the Depoissonnisation process, which is central in various analyses of the AofA domain. We first recall the two possible paths that may be used in this process. The first path, called here the Depoissonisation path, is better studied and is proven to apply in any practical situation; however, it often uses technical tools, that are not so easy to deal with. Moreover, the various results are scattered in the litterature, and the most recent results are not well known within the AofA domain. The present paper gathers in Section 2 all these results in a survey style. The second path, called here the Rice-Mellin path, is less often used within the AofA domain. It is often very easy to apply, but it needs a tameness condition, which appears {\em a priori} to be quite restrictive, and is not deeply studiedin the litterature. In Section 3, the paper precisely describes the Rice-Mellin path, together with its tameness condition, in a survey style, too. Finally, in Section 4, the paper presents original results for the Rice-Mellin path: it exhibits a framework, of practical use, where the tameness condition is proven to hold. It then proves that the Rice-Mellin path is both of easy and practical use : even though (much?) less general than the Depoissonisation path, it is easier to apply.