Moment-type estimates with an improved structure for the accuracy of the normal approximation to distributions of sums of independent symmetric random variables

摘要

For the uniform distance Δn between the distribution function of the standard normal law and the distribution function of the normalized sum of n independent random variables X1Xn with symmetric distribution functions F1,..., Fn and E Xj1,j EX2 j = σ 2 j, ∑n j =1,...,n, for all n ≧ 1 and c ≧ 0, the estimates x=supx>0(cosx-1+x2/2)/x3 =0.0991...,thefunctionA(c)isunboundedasc→0anddecreases monotonically, takes finite values for all c > 0, and is specified in the√xplicit form, the symbol ∧ standing for the minimum. The optimality of the constant (1/2 + x)/ 2π = 0.2390... in the first term corresponding to c = 0 is discussed. The obtained estimates improve the results of Bentkus [Preprint 91-078, Universit?t Bielefeld, Bielefeld, 1991; J. Theoret. Probab., 7 (1994), pp. 211-227] (with c = x) and Chistyakov [J. Math. Sci. (New York), 93 (1999), pp. 480-483; Theory Probab. Appl., 46 (2002), pp. 226-242; 516-522; 47 (2003), pp. 395-414] (with c = 1/2 - x = 0.4008...). Similar results are also proved for the case where symmetric random summands possess the absolute moments only of order 2 + δ with some 0 < δ < 1.