A fast Berry-Esseen theorem under minimal density assumptions

2023/05/29 by Johnston, Samuel G. G. · 2 citations
#60E15 #FOS: Mathematics #Primary: 60F05. Secondary: 60E10 #Probability (math.PR)

paper · doi:10.48550/arxiv.2305.18138

Abstract

Let X₁,…,X_N be i.i.d. random variables distributed like X. Suppose that the first k ≥ 3 moments \ 𝔼[X^j] : j = 1,…,k\ of X agree with that of the standard Gaussian distribution, that 𝔼[|X|^k+1] < ∞, and that there is a subinterval of ℝ of width w over which the law of X has a density of at least h. Then we show that sup_{s ∈ ℝ} | ℙ ( (X₁ + … + X_N)/( √(N) ) ≤ s ) - ∫_-∞^s \frac e ^{- u2/2} d u √(2 π) | ≤ 3 \ \frac𝔼[|X|^k+1] N ^(k-1)/(2) + e^ - c hw³ N/𝔼[|X|^k+1] \, where c > 0 is universal. By setting k=3, we see that in particular all symmetric random variables with densities and finite fourth moment satisfy a Berry-Esseen inequality with a bound of the order 1/N. Thereafter, we study the Berry-Esseen theorem as it pertains to perturbations of the Bernoulli law with a small density component, showing by means of a reverse inequality that the power hw³ in the exponential term is asymptotically sharp.

Cited by

• Fast mixing in Ising models with a negative spectral outlier via Gaussian approximation
• On the Optimality of the Median-of-Means Estimator under Adversarial Contamination

Related

• On rates of convergence and Berry-Esseen bounds for random sums of centered random variables with finite third moments
• An optimal Berry-Esseen type theorem for integrals of smooth functions
• A Berry-Esseen bound with (almost) sharp dependence conditions
• The Berry-Esseen bound in de Jong's CLT
• Berry-Esseen bounds in the entropic central limit theorem