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Nonasymptotic and distribution-uniform Komlós-Major-Tusnády approximation

Ian Waudby-Smith, Martin Larsson, Aaditya Ramdas

TL;DR

The paper develops distribution-uniform strong Gaussian approximations for sums of i.i.d. mean-zero variables, extending classical KMT results to entire families of distributions. It introduces Sakhanenko regularity and uniform moment conditions as necessary and sufficient criteria for uniform KMT approximations, providing explicit, nonasymptotic concentration inequalities that underpin the uniform couplings. The work treats two regimes: finite exponential moments (uniformly controlled) and finite power moments with uniform $q$-th integrability, establishing rates $O(\log n)$ and $o(n^{1/q})$ respectively. By instantiating the uniform results on a single probability space, it recovers the classical KMT-type conclusions while enabling distribution-robust applications and explicit constant control.

Abstract

We present nonasymptotic concentration inequalities for sums of independent and identically distributed random variables that yield asymptotic strong Gaussian approximations of Komlós, Major, and Tusnády (KMT) [1975,1976]. The constants appearing in our inequalities are either universal or explicit, and thus as corollaries, they imply distribution-uniform generalizations of the aforementioned KMT approximations. In particular, it is shown that uniform integrability of a random variable's $q^{\text{th}}$ moment is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $o(n^{1/q})$ for $q > 2$ and that having a uniformly lower bounded Sakhanenko parameter -- equivalently, a uniformly upper-bounded Bernstein parameter -- is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $O(\log n)$. Instantiating these uniform results for a single probability space yields the analogous results of KMT exactly.

Nonasymptotic and distribution-uniform Komlós-Major-Tusnády approximation

TL;DR

The paper develops distribution-uniform strong Gaussian approximations for sums of i.i.d. mean-zero variables, extending classical KMT results to entire families of distributions. It introduces Sakhanenko regularity and uniform moment conditions as necessary and sufficient criteria for uniform KMT approximations, providing explicit, nonasymptotic concentration inequalities that underpin the uniform couplings. The work treats two regimes: finite exponential moments (uniformly controlled) and finite power moments with uniform -th integrability, establishing rates and respectively. By instantiating the uniform results on a single probability space, it recovers the classical KMT-type conclusions while enabling distribution-robust applications and explicit constant control.

Abstract

We present nonasymptotic concentration inequalities for sums of independent and identically distributed random variables that yield asymptotic strong Gaussian approximations of Komlós, Major, and Tusnády (KMT) [1975,1976]. The constants appearing in our inequalities are either universal or explicit, and thus as corollaries, they imply distribution-uniform generalizations of the aforementioned KMT approximations. In particular, it is shown that uniform integrability of a random variable's moment is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of for and that having a uniformly lower bounded Sakhanenko parameter -- equivalently, a uniformly upper-bounded Bernstein parameter -- is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of . Instantiating these uniform results for a single probability space yields the analogous results of KMT exactly.

Paper Structure

This paper contains 34 sections, 19 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Preliminaries, notation, and uniform constructions
  3. Outline and summary of contributions
  4. Random variables with finite exponential moments
  5. Random variables with finite power moments
  6. Proofs from \ref{['section:exponential-moments']}
  7. Proof of \ref{['proposition:sakhanenko-param-bernstein-uniform-integrability']}
  8. Showing that $(i) \implies (iii)$
  9. Showing that $(iii) \implies (iv)$ and the inequality given in $(d)$
  10. Showing that $(iv) \implies (i)$
  11. Showing that $(i)$ and $(ii)$ are equivalent
  12. Proof of \ref{['theorem:concentration-kmt']}
  13. Step 1: Applying Sakhanenko's inequality at doubly exponential demarcations
  14. Step 2: Constructing the couplings on a single probability space
  15. Step 3: Partitioning $\mathbb N$ into doubly exponentially spaced epochs
  16. ...and 19 more sections

Key Result

Proposition 2.2

Let $X$ be a random variable with mean zero on the collection of probability spaces $(\Omega_\alpha, \mathcal{F}_\alpha, P_{\alpha})_{\alpha \in \mathcal{A}}$ with uniformly positive variance: $\inf_{\alpha \in \mathcal{A}} \mathrm{Var}_{P_\alpha}(X) \geq \underaccent{\bar{}}{\sigma}^2 > 0$. The fol Furthermore, we have the following relations between the constants above:

Theorems & Definitions (39)

  • Definition 2.1: Sakhanenko regularity
  • Proposition 2.2: Equivalent characterizations of Sakhanenko regularity
  • Theorem 2.3: Distribution-uniform Komlós-Major-Tusnády approximation
  • Theorem 2.4: Nonasymptotic Komlós-Major-Tusnády approximation with finite exponential moments
  • Definition 3.1: Distribution-uniform strongly approximated processes
  • Theorem 3.2: Distribution-uniform Komlós-Major-Tusnády approximation for finite power moments
  • Theorem 3.3: Nonasymptotic strong approximation with finite power moments
  • Corollary 3.4
  • proof : Proof of \ref{['proposition:sakhanenko-param-bernstein-uniform-integrability']}
  • Lemma 4.1: Bounds on polynomial moments from exponential ones
  • ...and 29 more