假設 X1,…,Xn 是 n 個獨立且分配相同的隨機變數，而 X1n,X2n,…,Xnn 是其有序統計量，令Tn=n^(-1)*( i=1,2,...,n時, J(i/n)*h(Xin) 的和 ) 這樣形式的函數組合是有序統計量的線性組合(簡稱L-統計量 )。四十多年來一直有眾多學者研究L-統計量。
在這篇文章中，我們感興趣的是在適當的條件下找尋此統計量趨向常態分配的誤差界限，同時在適當的可測函數 J 與 h 上，建立了貝瑞伊申界的收斂速度n^(-(1/2))*logn 。

Let X1,…,Xn be n independent and identically distributed random variables , X1n,X2n,…,Xnn be their corresponding order statistics , and Tn=n^(-1)*( the sum of J(i/n)*h(Xin), i=1,2,...n) be the corresponding linear combination of order statistics ( L-statistics ) . L-statistics have received much attention during the last forty years .
In this paper , we are interested in the rate of convergence to normality under suitable conditions , and establish a Berry-Essen bound of order n^(-(1/2))*logn for Tn under some regular conditions on the functions J and h .

References
[1] S. Bjerve . Error bounds for linear combinations of order statistics . Ann.
Statist. , V.5 , No.1-2 , p.357-369 , 1977 .
[2] H. Chernoff , J. L. Gastwirth and M. V. Johns , Jr. . Asymptotic distribution
of linear combinations of functions of order statistics with applications to
estimation . Ann. Math. Statist. , 38 , p.52-72 , 1967 .
[3] Kai Lai Chung . A Course in Probability Theory . Harcourt Brace
Jovanovich , Inc. , p.225 , 1968 .
[4] M. Csorgo and , P. Revesz . Strong Approximations in Probability and
Statistics . Academic Press , Inc. , p.128 , 1981 .
[5] H. A. David . Order Statistics . Wiley , New York , p.93 , 1969 .
[6] B. V. Gnedenko and A. N. Kolmogorov . Limit Distributions for Sums of
Independent Random Variables (translated from the Russian) . Addison-
Wesley Publishing Co. Inc. Reading , Mass. , p.201 , 1954 .
[7] R. Helmers . A Berry-Essen theorem for linear combination of order
statistics . Ann. Prob. V.9 , No.2 , p.342-347 , 1981 .
[8] Cheun Der Lea and Madan L. . Asymptotic properties of linear functions of
order statistics . J. Statist. Planning . And Infer. , 18 , p.203-223 , 1988 .
[9] Deli Li , M. B. Rao and R. J. Tomkins . The law of the iterated logarithm and
central limit theorem for L-statistics . J. Mult. Anal. , 78(2) , p.191-217 ,
2001 .
[10] D. Mason and G. Shorack . Necessary and sufficient conditions for
asymptotic normally of L-statistics . Ann. Probab. 20(4) , p.1779-1804 ,
1992 .
[11] D. S. Moore . An elementary proof of asmptotic normality of linear
functions of order statistics . Ann. Math. Statist. , 39 , part 1 ,
p.263-265 , 1968 .
[12] V. V. Petrov . Sums of Independent Random Variables (translated from
the Russian) . Springer-Verlag , p.286 , 1975 .
[13] W. Rosenkrantz and N. E. O’Reilly . Application of the Skorokhod
representation theorem to rates of convergence for linear combinations
of order statistics . Ann. Math. Statist. , 43 , part 2 , p.1204-1212 , 1972 .
[14] R. J. Serfing . Approximation Theorems of Mathematical Statistics .
Wiley , New York , p.262 , 1980 .
[15] R. J. Serfling . Generalized L- , M- , R-Statistics . Ann. Statist. , 12 ,
p.76-86 , 1984 .
[16] S. M. Stigler . Linear functions of order statistics . Ann. Math. Statist. ,
40 , No.1-3 , p.770-788 , 1969 .
[17] S. M. Stigler . Linear functions of order statistics with smooth weight
functions . Ann. Statist. , 2 , No.4-6 , p.676-693 , 1974 .
[18] T. De Wet . Rate of convergence of linear combinations of order
statistics . S. Afric. Statist. J. , 8 , p.35-43 , 1974 .