Berry–Esseen Bounds for Independent Sums

1. Nonuniform Berry--Esseen bounds🔗

Let \Phi(x)=\Pr(Z\le x) for a standard normal random variable Z. In this section I is finite, (\Omega,\mathcal F,\mathbb P) is a probability space, and (X_i)_{i\in I} is an independent measurable real-valued family. We write W=\sum_iX_i and \gamma=\sum_i\mathbb E|X_i|^3 whenever the moments exist.

The proof proceeds from the half-line Stein equation through concentration, one-sided truncation, and an exact residual decomposition. The final reflection step converts the positive-threshold estimate into a bound on the whole real line.

  1. 1.1. The half-line Stein equation
  2. 1.2. Uniform approximation and leave-one-out concentration
  3. 1.3. Exponential concentration
  4. 1.4. One-sided truncation
  5. 1.5. Stein exchange and residual decomposition
  6. 1.6. Estimates for the residual terms
  7. 1.7. The central nonuniform estimate
  8. 1.8. Reflection and the nonuniform theorem