A central limit theorem for decomposable random variables with applications to random graphs

Normal approximation for random sums

“…It readily delivers error bounds which are often of or close to the correct asymptotic order, when the distance between distributions is measured with respect to the (bounded) Wasserstein distance; see, for example, Erickson (1974) and Barbour et al (1989). If a bound for the error in Kolmogorov distance, d K , is preferred (where, for two probability measures P and Q on R, d K (P , Q) := sup x |P (−∞, x] − Q(−∞, x]|), the arguments needed are more involved, but there have nonetheless been notable successes, such as Bolthausen's (1984) Berry-Esseen bound for the combinatorial central limit theorem.…”

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“…In Barbour et al 8 the authors explicitly restricted their considerations to smooth test functions but Raič 19 used the Lipschitz test function to modify the proof of Barbour et al 8 for non-smooth test functions at the cost of a boundedness condition.…”

“…In order to use (8) to obtain the bound of normal approximation, many authors [20][21][22] , choose the test function h = I z where…”

Symmetric and centered binomial approximation of sums of locally dependent random variables

“…The proof is omitted, because it runs analogously to the proof of Theorem 3.1; see also Barbour et al [6].…”

“…We exploit a finite local dependence structure as presented in Chen and Shao [12]. In the context of Stein's method for normal approximation, it has been successfully applied to a variety of problems; see for example Stein [31] (Lecture 10), Barbour et al [6], Rinott and Rotar [26], Dembo and Rinott [15] and Barbour and Xia [3]. Note that Barbour et al [6] use a slightly more general dependence structure, often yielding crucial improvement when approximating sums of dissociated random variables by the normal distribution.…”