Laws of large numbers for L-statistics. (1994) - Norvaiša, Rimas
Abstract : Consider L n = n − 1 ∑ 1 ≤ i ≤ n c n i g ( X n : i ) for order statistics X n : i and let c n i = n ∫ ( i − 1 ) / n i / n J d λ for some (Lebesgue) λ -summable over ( 0, 1 ) function J . Sufficient as well as necessary conditions for lim n L n = ∫ 0 1 J g d λ to hold almost surely and in probability are given. Superposition (or Nemytskii) operators have been used to derive the laws of large numbers for L -statistics from the laws of large numbers in quasi-Banach function spaces for the empirical distribution functions based on X 1, …, X n .