Moment-type estimates with asymptotically optimal structure for the accuracy of the normal approximation

For the uniform distance Delta(n) between the distribution function of the standard normal law and the distribution function of the standardized sum of independent random variables X-1, ... , X-n with EXj = 0, E vertical bar X-j vertical bar = beta(1, j), EXj2 = sigma(2)(j), j = 1, ... , n, for all n >= 1 the bounds Delta(n) <= 2l(n)/3 root 2 pi +1/2 root 2 pi B-n(3) Sigma(n)(j=1) beta(1,j) sigma(2)(j) + R(l(n)), Delta(n) <= inf (c >= 2/(3 root 2 pi)) {cl(n) + K(c)/B-n(3) Sigma(n)(j=1) sigma(3)(j) + R-c(l(n))}, are proved, where B-n(2) = Sigma(n)(j=1) sigma(2)(j), l(n) = B-n(-3) Sigma(n)(j=1) E vertical bar X-j vertical bar(3), R(l(n)) <= 6l(n)(5/3), R-c(l(n)) <= min{3l(n)(7/6), A(c)l(n)(4/3)} in the general case and R(l(n)) <= 3l(n)(2), R-c(l(n)) <= min {2l(n)(3/2) , A(c)l(n)(2)}, if X-1, ... , X-n are identically distributed, A(c) > 0 being a decreasing function of c such that A(c) -> infinity as c -> 2/(3 root 2 pi). More-over, the function K(c) is optimal for each c >= 2/(3 root 2 pi). In particular, K ((root 10+3)/(6 root 2 pi)) = 0, K (2/3 root 2 pi)) = root(2 root 3-3)/(6 pi) = 0.1569 ... It is shown that in the first inequality the coefficients 2/(3 root 2 pi) and (2 root 2 pi)(-1) are optimal and the lower bound 2/(3 root 2 pi) for c in the second inequality is unimprovable. These results sharpen the well-known estimates due to H. Prawitz (1975), V. Bentkus (1991, 1994) and G. P. Chistyakov (1996, 2001). Also, an analog of the first inequality is proved for the case where the summands possess only the moments of order 2 vertical bar delta with some 0 < delta < 1. As a by-product, the von Mises inequality for lattice distributions is sharpened and generalized.