@article{oai:ynu.repo.nii.ac.jp:00006804,
 author = {Fotopoulos, S. B.},
 issue = {1},
 journal = {Yokohama Mathematical Journal = 横濱市立大學紀要. D部門, 数学},
 month = {},
 note = {application/pdf, Let ＄￥{X_{n}, n￥in N￥}＄ be a sequence of independent random variables with ＄E[X_{n}]=0＄ and ＄ E[X_{n}^{2}]<￥infty＄ , and let ＄￥{N_{n}, n￥in N￥}＄ be a sequence of random indices defined on the same space as ＄X_{n}' s＄ and independent of the latter. We write ＄S_{N_{n}}=￥sum_{i=}^{N_{n_{1}}}X_{i}＄ and ＄ L_{n}^{2}=Var(S_{N_{n}})<￥infty＄ . We show that for the validity ＄￥lim_{n￥rightarrow￥infty}￥Vert(1+|x|)^{2-1/p}|P(S_{N_{n}}￥leq L_{n}x)-￥Phi(x)|￥Vert_{p}=0＄ it is required that ￥lim_{n￥rightarrow￥infty}L_{n}^{-2-￥delta}E[￥Sigma_{i=1}^{N_{n}}E[|X_{i}|^{2+￥delta}]=0＄ for ＄ 1￥leq P￥leq￥infty＄ and ＄￥delta￥in(0,1)＄ , and ＄￥lim_{n￥rightarrow￥infty}L_{n}^{2-2}＄ . Var ＄(s_{N_{n}}^{2})^{1/2}=0＄ . Furthermore, we provide conditions under which the series ＄[￥Sigma_{n=1}^{￥infty}(g(L_{n})￥Vert(1+|x|)^{2-1/p}|P(S_{N_{n}}￥leq L_{n}x)-Phi(x)|￥Vert_{p})^{S}(t_{n}^{2}/L_{n}^{2})]^{1/S}＄ converges for any ＄ 1￥leq s<￥infty＄ .},
 pages = {61--76},
 title = {NON-UNIFORM ESTIMATES IN THE CENTRAL LIMIT THEOREM FOR RANDOM SUMS OF INDEPENDENT RANDOM VARIABLES},
 volume = {37},
 year = {1989}
}