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Dey, Partha S.; Terlov, Grigory

Stein’s method for conditional central limit theorem. (English) Zbl 1514.60035

Ann. Probab. 51, No. 2, 723-773 (2023).
The authors extend Stein’s method to give error bounds and rates of convergence in univariate and multivariate conditional central limit theorems. That is, they give explicit error bounds (measured in 1-Wasserstein distance) in Gaussian approximation for random variables of the form \((W|Y=k)\), and analogous multivariate results. Here \(Y\) is assumed to be a discrete random variable uncorrelated with \(W\). The approach the authors take is motivated by the well-known exchangeable pairs approach to the usual central limit theorem via Stein’s method, and uses linearity conditions of a type familiar from that approach. Applications given include the appearance of patterns in random binary sequences, and subgraph counts in an Erdos-Rényi random graph. The paper concludes with a discussion of open problems and potential future work.
Reviewer: Fraser Daly (Edinburgh)

MSC:

60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
05C80 Random graphs (graph-theoretic aspects)
62E17 Approximations to statistical distributions (nonasymptotic)

Keywords:

central limit theorem; conditional law; multivariate normal approximation; rate of convergence; Stein’s method
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References:

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