Abstract
The general results of Chapters 3 and 6 can be applied in many contexts, and the main difficulty is then to prove for each case that one has indeed the estimate on the Laplace transform given by Definition 1.1.1. Therefore, the development of techniques to obtain mod-φ estimates is an important part of the work. Such an estimate can sometimes be established from an explicit expression of the Laplace transform (hence of the characteristic function); we give several examples of this kind in Section 7.1. But there also exist numerous techniques to study sequences of random variables without explicit expression for the characteristic function: complex analysis methods in number theory (Section 7.2) and in combinatorics (Section 7.3), localisation of zeros (Chapter 8) and dependency graphs (Chapters 9, 10 and 11) to name a few. These methods are known to yield central limit theorems and we show how they can be adapted to prove mod-convergence. We illustrate each case with one or several example(s).
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Féray, V., Méliot, PL., Nikeghbali, A. (2016). Examples with an explicit generating function. In: Mod-ϕ Convergence. SpringerBriefs in Probability and Mathematical Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-46822-8_7
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DOI: https://doi.org/10.1007/978-3-319-46822-8_7
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