[Paper Review] Estimates for the strong approximation in multidimensional central limit theorem

A. Yu. Zaîtsev|ArXiv.org|Apr 24, 2003

Probability and Risk Models29 references32 citations

TL;DR

This paper establishes explicit, dimension-dependent bounds for strong Gaussian approximation of sums of independent d-dimensional random vectors with finite exponential moments. Using a dyadic construction and quantile transforms, it generalizes Komlós–Major–Tusnády and Sakhanenko's one-dimensional results to the multidimensional case, providing explicit dependence on dimension d and distributional parameters via exponential tail bounds for the approximation error.

ABSTRACT

In a recent paper the author obtained optimal bounds for the strong Gaussian approximation of sums of independent $\R^d$-valued random vectors with finite exponential moments. The results may be considered as generalizations of well-known results of Komlós--Major--Tusnády and Sakhanenko. The dependence of constants on the dimension $d$ and on distributions of summands is given explicitly. Some related problems are discussed.

Motivation & Objective

• To extend optimal strong approximation results from one-dimensional to multidimensional settings for sums of independent random vectors.
• To provide explicit dependence of approximation constants on dimension d and on the tail behavior of summand distributions.
• To generalize Sakhanenko's and Komlós–Major–Tusnády's one-dimensional results to d-dimensional random vectors with finite exponential moments.
• To establish sharp tail bounds for the Prokhorov distance and the approximation error in the multidimensional invariance principle.
• To address open questions regarding the necessity of covariance structure conditions in high-dimensional strong approximation.

Proposed method

• Adapts the dyadic approximation scheme of Komlós–Major–Tusnády to the multidimensional case using conditional quantile transforms.
• Constructs independent Gaussian vectors $Y_1, \dots, Y_n$ with matching first and second moments to the original $X_i$.
• Uses smooth, bounded approximating distributions with matching first, second, and third moments to control approximation error.
• Employs the Strassen–Dudley theorem to couple the original sum with a Gaussian process via Prokhorov distance estimates.
• Applies the Rosenblatt quantile transform for conditional distributions in the dyadic block construction to ensure smoothness and closeness to Gaussian.
• Derives exponential tail bounds for the maximum deviation $\Delta(X,Y) = \max_{1 \leq k \leq n} \|\sum_{i=1}^k (X_i - Y_i)\|$ using properties of the class $\mathcal{A}_d(\tau)$.

Experimental results

Research questions

• RQ1 What are the optimal explicit bounds for the strong approximation error in the multidimensional central limit theorem?
• RQ2 How does the approximation error depend on the dimension $d$ and the tail behavior of the summand distributions?
• RQ3 Can the dyadic construction method be extended to non-identically distributed, d-dimensional random vectors with finite exponential moments?
• RQ4 What is the role of the covariance structure and the parameter $\tau$ in determining the approximation rate?
• RQ5 Is the condition on the covariance operator (e.g., $\text{cov}(F) \geq \tau \mathbf{I}$) necessary for the strong approximation bounds?

Key findings

• The paper establishes an exponential tail bound: $\mathbf{E}\left[\exp\left(\frac{c \Delta(X,Y)}{\tau}\right)\right] \leq 1 + B/\tau$, where $B^2 = \sum_{i=1}^n \mathbf{E}[X_i^2]$, with explicit constants depending on dimension and distribution.
• For the Prokhorov distance, the bound $\pi(F, \Phi(F), \lambda) \leq c d^2 \exp\left(-\frac{\lambda}{c d^2 \tau}\right)$ holds for all $\tau > 0$, even without the covariance condition.
• The approximation error satisfies $\mathbf{P}(c_1 \Delta(X,Y)/\tau(F) \geq x) \leq \exp\left(\log(1 + \sqrt{n \mathbf{E}[\xi^2]}/\tau(F)) - x\right)$, explicitly showing dependence on the distribution $F$.
• For convolutions of distributions in $\mathcal{B}_d(\tau)$, the coupling satisfies $\mathbf{P}(\|\xi - \eta\| > \lambda) \leq c(d)\left(\max_i p_i + \exp(-\lambda / (c(d)\tau))\right) + \sum p_i^2$, generalizing to unbounded supports.
• The method removes a logarithmic factor present in earlier multidimensional results, achieving optimality in the sense of KMT and Sakhanenko.
• The construction is valid for non-identically distributed summands and provides a sharp, dimension-aware approximation rate with explicit constants.