Optimal constant in the Kolmogorov-distance normal approximation for Bernoulli convolutions (including binomial and Poisson laws)

Determine the optimal universal constant c_K such that for every distribution obtained as a weak limit of sums of independent Bernoulli random variables with possibly varying success probabilities (i.e., a Bernoulli convolution or a weak limit thereof), with mean μ and standard deviation σ > 0, the Kolmogorov distance between its standardised distribution function F(x) = P((X−μ)/σ ≤ x) and the standard normal distribution function Φ satisfies sup_x |F(x) − Φ(x)| ≤ c_K/σ. Ascertain whether the asymptotically motivated conjectured value proposed by the authors indeed equals the true optimal constant.

Background

Remark 1.2(b) provides a uniform bound of the form |F − Φ|_∞ ≤ min{0.5410, 0.3440/σ} for the Kolmogorov distance between the standardised law and the normal distribution. Remark 1.2(e) states that this extends to the class of Bernoulli convolutions and their weak limits.

Remark 1.2(f) explicitly notes that the constant 0.3440 is most likely not optimal and points to natural conjectures for the true optimal constant. In the symmetric binomial/hypergeometric case the bound improves to 1/(2√(2π)) ≈ 0.1994, indicating room for improvement in the general case. The open problem is to identify the exact optimal constant c_K and assess whether the conjectured value is correct.