Stein’s method in high dimensions with applications

Adrian Röllin

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 2, page 529-549
  • ISSN: 0246-0203

Abstract

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Let h be a three times partially differentiable function on ℝ n , let X = ( X 1 , ... , X n ) be a collection of real-valued random variables and let Z = ( Z 1 , ... , Z n ) be a multivariate Gaussian vector. In this article, we develop Stein’s method to give error bounds on the difference 𝔼 h ( X ) - 𝔼 h ( Z ) in cases where the coordinates of X are not necessarily independent, focusing on the high dimensional case n → ∞ . In order to express the dependency structure we use Stein couplings, which allows for a broad range of applications, such as classic occupancy, local dependence, Curie–Weiss model, etc. We will also give applications to the Sherrington–Kirkpatrick model and last passage percolation on thin rectangles.

How to cite

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Röllin, Adrian. "Stein’s method in high dimensions with applications." Annales de l'I.H.P. Probabilités et statistiques 49.2 (2013): 529-549. <http://eudml.org/doc/271945>.

@article{Röllin2013,
abstract = {Let $h$ be a three times partially differentiable function on $\mathbb \{R\}^\{n\}$, let $X=(X_\{1\},\ldots ,X_\{n\})$ be a collection of real-valued random variables and let $Z=(Z_\{1\},\ldots ,Z_\{n\})$ be a multivariate Gaussian vector. In this article, we develop Stein’s method to give error bounds on the difference $\mathbb \{E\}h(X)-\mathbb \{E\}h(Z)$ in cases where the coordinates of $X$ are not necessarily independent, focusing on the high dimensional case $n\rightarrow \infty $. In order to express the dependency structure we use Stein couplings, which allows for a broad range of applications, such as classic occupancy, local dependence, Curie–Weiss model, etc. We will also give applications to the Sherrington–Kirkpatrick model and last passage percolation on thin rectangles.},
author = {Röllin, Adrian},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Stein’s method; gaussian interpolation; last passage percolation on thin rectangles; Sherrington–Kirkpatrick model; Curie–Weiss model; Stein's method; Gaussian interpolation; last passage percolation; Sherrington-Kirkpatrick model; Curie-Weiss model},
language = {eng},
number = {2},
pages = {529-549},
publisher = {Gauthier-Villars},
title = {Stein’s method in high dimensions with applications},
url = {http://eudml.org/doc/271945},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Röllin, Adrian
TI - Stein’s method in high dimensions with applications
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 2
SP - 529
EP - 549
AB - Let $h$ be a three times partially differentiable function on $\mathbb {R}^{n}$, let $X=(X_{1},\ldots ,X_{n})$ be a collection of real-valued random variables and let $Z=(Z_{1},\ldots ,Z_{n})$ be a multivariate Gaussian vector. In this article, we develop Stein’s method to give error bounds on the difference $\mathbb {E}h(X)-\mathbb {E}h(Z)$ in cases where the coordinates of $X$ are not necessarily independent, focusing on the high dimensional case $n\rightarrow \infty $. In order to express the dependency structure we use Stein couplings, which allows for a broad range of applications, such as classic occupancy, local dependence, Curie–Weiss model, etc. We will also give applications to the Sherrington–Kirkpatrick model and last passage percolation on thin rectangles.
LA - eng
KW - Stein’s method; gaussian interpolation; last passage percolation on thin rectangles; Sherrington–Kirkpatrick model; Curie–Weiss model; Stein's method; Gaussian interpolation; last passage percolation; Sherrington-Kirkpatrick model; Curie-Weiss model
UR - http://eudml.org/doc/271945
ER -

References

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