# A high-dimensional CLT in $$\mathcal {W}_2$$ W 2 distance with near optimal convergence rate

A high-dimensional CLT in $$\mathcal {W}_2$$ W 2 distance with near optimal convergence rate Let $$X_1,\ldots ,X_n$$ X 1 , … , X n be i.i.d. random vectors in $$\mathbb {R}^d$$ R d with $$\Vert X_1\Vert \le \beta$$ ‖ X 1 ‖ ≤ β . Then, we show that \begin{aligned} \frac{1}{\sqrt{n}}\left( X_1 + \cdots + X_n\right) \end{aligned} 1 n X 1 + ⋯ + X n converges to a Gaussian in quadratic transportation (also known as “Kantorovich” or “Wasserstein”) distance at a rate of $$O \left( \frac{\sqrt{d} \beta \log n}{\sqrt{n}} \right)$$ O d β log n n , improving a result of Valiant and Valiant. The main feature of our theorem is that the rate of convergence is within $$\log n$$ log n of optimal for $$n, d \rightarrow \infty$$ n , d → ∞ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Probability Theory and Related Fields Springer Journals

# A high-dimensional CLT in $$\mathcal {W}_2$$ W 2 distance with near optimal convergence rate

, Volume 170 (4) – Mar 24, 2017
25 pages

/lp/springer_journal/a-high-dimensional-clt-in-mathcal-w-2-w-2-distance-with-near-optimal-565OACu0t8
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Probability Theory and Stochastic Processes; Theoretical, Mathematical and Computational Physics; Quantitative Finance; Mathematical and Computational Biology; Statistics for Business/Economics/Mathematical Finance/Insurance; Operations Research/Decision Theory
ISSN
0178-8051
eISSN
1432-2064
D.O.I.
10.1007/s00440-017-0771-3
Publisher site
See Article on Publisher Site

### Abstract

Let $$X_1,\ldots ,X_n$$ X 1 , … , X n be i.i.d. random vectors in $$\mathbb {R}^d$$ R d with $$\Vert X_1\Vert \le \beta$$ ‖ X 1 ‖ ≤ β . Then, we show that \begin{aligned} \frac{1}{\sqrt{n}}\left( X_1 + \cdots + X_n\right) \end{aligned} 1 n X 1 + ⋯ + X n converges to a Gaussian in quadratic transportation (also known as “Kantorovich” or “Wasserstein”) distance at a rate of $$O \left( \frac{\sqrt{d} \beta \log n}{\sqrt{n}} \right)$$ O d β log n n , improving a result of Valiant and Valiant. The main feature of our theorem is that the rate of convergence is within $$\log n$$ log n of optimal for $$n, d \rightarrow \infty$$ n , d → ∞ .

### Journal

Probability Theory and Related FieldsSpringer Journals

Published: Mar 24, 2017

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