The sharp form of the Kolmogorov--Rogozin inequality and a conjecture of Leader--Radcliffe

2022/01/24 by Tomas Juškevičius, Juškevičius, Tomas · 2 citations
Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Fuzzy Systems and Optimization #Mathematical Dynamics and Fractals #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2201.09861

openalex publication_date 2022/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a random variable and define its concentration function by Q_h(X)=sup_{x∈ ℝ}ℙ(X∈ (x,x+h]). For a sum S_n=X₁+⋯+X_n of independent real-valued random variables the Kolmogorov-Rogozin inequality states that Q_h(S_n)≤ C(∑_i=1ⁿ(1-Q_h(X_i)))^-(1)/(2). In this paper we give an optimal bound for Q_h(S_n) in terms of Q_h(X_i), which settles a question posed by Leader and Radcliffe in 1994. Moreover, we show that the extremal distributions are mixtures of two uniform distributions each lying on an arithmetic progression.

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