A note to the convergence rates in precise asymptotics

Let {X, X-n, n >= 1} be a sequence of i. i. d. random variables with zero mean. Set S-n = Sigma(n)(k=1) X-k, EX2 = sigma(2) > 0, and lambda(r,p)(epsilon) = Sigma(infinity)(n=1) n(r)/(p)-P-2(vertical bar S-n vertical bar >= n(1/p) epsilon). In this paper, the author discusses the rate of approximation of p/r-p E vertical bar N vertical bar(2(r-p)/(2-p)) by epsilon(2(r-p)/(2-p))lambda(r,p)(epsilon) under suitable moment conditions, where N is normal with zero mean and variance sigma(2) > 0, which improves the results of Gut and Steinebach ( J. Math. Anal. Appl. 390: 1- 14, 2012) and extends the work He and Xie ( Acta Math. Appl. Sin. 29: 179- 186, 2013). Specially, for the case r = 2 and p = 1/beta+1, beta > - 1/2, the author discusses the rate of approximation of sigma(2)/2 beta+1 by epsilon(2)lambda(2,1/(beta+1))(epsilon) under the condition EX(2)l(vertical bar X vertical bar > t) = O(t(-delta)l(t)) for some delta > 0, where l(t) is a slowly varying function at infinity. MSC: 60F15; 60G50