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

Abstract Let be a random variable and define its concentration function by For a sum of independent real‐valued random variables, the Kolmogorov–Rogozin inequality states that In this paper, we give an optimal bound for in terms of , 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.