This chapter derives some probabilistic results for later use. A dependency graph for a family of random variables is a deterministic graph with vertex-set identified with the random variables and edges corresponding to dependencies between variables. Univariate and multivariate Poisson approximation theorems, and normal approximation theorems, are derived for sums of variables having a dependency graph using Stein's method. Random variables given as sums of martingale differences are also considered. For such variables, both concentration inequalities and a central limit theorem are given. Finally, a ‘de-Poissonization’ result is given which is to be used for extending central limit theorems for G(N(n),r) (with N(n) Poisson with mean n) to analogous results for G(n,r).
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