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

Determine the optimal universal constant c_I 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, and for every interval I = (a, b] (allowing endpoints ±∞), the interval-probability normal approximation error satisfies |P(I) − [Φ((b − μ)/σ) − Φ((a − μ)/σ)]| ≤ c_I/σ. Ascertain whether the asymptotically motivated conjectured value proposed by the authors indeed equals the true optimal constant.

Background

Theorem 1.1 establishes a simple, unified upper bound for normal approximation errors of interval probabilities for binomial, hypergeometric, and Poisson distributions—namely, an error of order 1/σ with a proven universal constant 0.6879. Remark 1.2(e) extends this bound to the broader class of distributions formed by limits of Bernoulli convolutions (which includes these three distribution families).

Remark 1.2(f) explicitly notes that the constant 0.6879 in this bound is most likely not optimal and mentions natural conjectures for the optimal constant based on asymptotic considerations. For symmetric cases (e.g., symmetric binomial or hypergeometric laws), the constant can be improved to 1/√(2π) ≈ 0.3989, underscoring that tighter constants are attainable in special settings. The open problem is to pin down the exact optimal constant c_I for the general (non-symmetric) case and to decide whether the conjectured value is indeed correct.