2022/11/04 by Stanisław Cichomski, Cichomski, Stanisław, Adam Osękowski +1
Mathematics · #Benford’s Law and Fraud Detection #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2211.02446
openalex publication_date 2022/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X1, X2, …, Xn be a sequence of coherent random variables, i.e., satisfying the equalities Xj=ℙ(A|Gj), j=1, 2, …, n, almost surely for some event A. The paper contains the proof of the estimate ℙ(max1≤ i lt; j≤ n|Xi-Xj|≥ δ) ≤ (n(1-δ))/(2-δ) \wedge 1, where δ∈ ((1)/(2),1] is a given parameter. The inequality is sharp: for any δ, the constant on the right cannot be replaced by any smaller number. The argument rests on several novel combinatorial and symmetrization arguments, combined with dynamic programming. Our result generalizes the two-variate inequality of K. Burdzy and S. Pal and in particular provides its alternative derivation.