The Stein-Tikhomirov method and Berry-Esseen inequality for sampling sums from a finite population of independent random variables

被引：0
作者：
机构：
[1] National University of Uzbekistan, Tashkent
[2] Tashkent Institute of Motor Car and Road Engineers, Tashkent
来源：
Formanov, S. K. (shakirformanov@yandex.ru) | 1600年 / Springer Science and Business Media, LLC卷 / 33期
关键词：
Berry-Esseen inequality; Characteristic function; Distribution function; Independentrandom variables; Sampling sums from finite population; Stein-Tikhomirov method;
D O I：
10.1007/978-3-642-33549-5_14
中图分类号：
学科分类号：
摘要：
We present a simplified version of the Stein-Tikhomirov method realized by defining a certain operator in class of twice differentiable characteristic functions. Using this method, we establish a criterion for the validity of a nonclassical central limit theorem in terms of characteristic functions, in obtaining of classical Berry-Esseen inequality for sampling sums from finite population of independent random variables. © Springer-Verlag Berlin Heidelberg 2013.
引用
收藏
页码：261 / 273
页数：12
相关论文
共 12 条
  • [1] Bikelis A., On the estimation of the remainder term in the central limit theorem for samples from finite populations, Studia Sci. Math. Hungar., 4, pp. 345-354, (1969)
  • [2] Bolthausen E., An estimate of the remainder in a combinatorial limit theorem, Z. Wahr. Verw. Geb., 66, pp. 379-386, (1984)
  • [3] Linnik Y.I., Ostrovskii I.V., Expansions of Random Variables and Vectors [in Russian]., (1972)
  • [4] Liptser R.S., Shiryaev A.N., Martingale Theory [in Russian]., (1986)
  • [5] Petrov V., Limit Theorems for Sums of Independent Random Variables [in Russian]., (1987)
  • [6] Rosen B., Limit theorems for sampling from finite populations, Arch. Math., 5, pp. 383-424, (1964)
  • [7] Rotar V.I., On the generalization of the Lindeberg-Feller theorem, Mat. Zametki [Math. Notes], 18, 1, pp. 129-135, (1975)
  • [8] Soo-Thong H., Chen L.H.Y., An Lp bound for the remainder in combinatorial central limit theorem, Ann. Probab., 6, 2, pp. 231-249, (1978)
  • [9] Stein C., A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, Proceedings of the Sixth Berkley Symposium on Mathematical Statistics and Probability, 2, pp. 583-602, (1972)
  • [10] Tikhornirov A.N., On the convergence of the remainder in the central limit theorem for weakly dependent random variables, Teor. Veroyatnost. i Primenen [Theory Probab. Appl.], 25, 4, pp. 800-818, (1980)
12