High-dimensional central limit theorems by Stein’s method

By Xiao Fang, Yuta Koike. The Annals of Applied Probability, 2020

51 citations recorded by the source corpus.

We obtain explicit error bounds for the $d$-dimensional normal approximation on hyperrectangles for a random vector that has a Stein kernel, or admits an exchangeable pair coupling, or is a non-linear statistic of independent random variables or a sum of $n$ locally dependent random vectors. We assume the approximating normal distribution has a non-singular covariance matrix. The error bounds vanish even when the dimension $d$ is much larger than the sample size $n$. We prove our main results using the approach of Gotze (1991) in Stein's method, together with modifications of an estimate of Anderson, Hall and Titterington (1998) and a smoothing inequality of Bhattacharya and Rao (1976). For sums of $n$ independent and identically distributed isotropic random vectors having a log-concave density, we obtain an error bound that is optimal up to a $\log n$ factor. We also discuss an application to multiple Wiener-Ito integrals.

Subfields: Applied Mathematics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1a315eb3d0701e1f75691950c647a3cf83ac3161

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Citations in the corpus

Nearly optimal central limit theorem and bootstrap approximations in high dimensions (2020, methodology): “[7, 28, 3, 8, 11, 36, 12, 15, 23, 32, 31], but the key technical tool behind our results is a set of new smoothing inequalities, which we call mixed smoothing inequalities.”
Improved central limit theorem and bootstrap approximations in high dimensions (2019, unknown): “In particular, Fang and Koike [23] showed that the right-hand side of (4) can be improved to C(log(4)(pn)/n)1/3 when the covariance matrices are non-degenerate and can be further improved to C(log(3)(p)/n)1/2 logn when we additionally assume that Xi’s have log-concave densities.”