Abstract
The theory of dependency graphs is a powerful toolbox to prove asymptotic normality. A dependency graph encodes the dependency structure in a family of random variables: roughly we take a vertex for each variable in the family and connect dependent random variables by edges. The idea is that, if the degrees in a sequence of dependency graphs do not grow too fast, then the corresponding variables behave as if independent and the sum of the corresponding variables is asymptotically normal. Precise normality criteria using dependency graphs have been given by Petrovskaya/Leontovich, Janson, Baldi/Rinott and Mikhailov [PL83, Jan88, BR89, Mik91]. These results are black boxes to prove asymptotic normality of sums of partially dependent variables and can be applied in many different contexts. The original motivation of Petrovskaya and Leontovich comes from the mathematical modelisation of cell populations [PL83]. On the contrary, Janson was interested in random graph theory: dependency graphs are used to prove central limit theorems for some statistics, such as subgraph counts, in G(n, p) [BR89, Jan88, JŁR00]; see also [Pen02] for applications to geometric random graphs. The theory has then found a field of application in geometric probability [AB93, PY05, BV07]. More recently it has been used to prove asymptotic normality of pattern counts in random permutations [Bón10, HJ10]. Dependency graphs also generalise the notion of m-dependence [HR48, Ber73], widely used in statistics [Das08].
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Féray, V., Méliot, PL., Nikeghbali, A. (2016). Dependency graphs and mod-Gaussian convergence. In: Mod-ϕ Convergence. SpringerBriefs in Probability and Mathematical Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-46822-8_9
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