**Sankhya: The Indian Journal of Statistics**

1996, Volume 58, Series A, Pt. 1, pp. 126--134

**AN L _{p} NONUNIFORM CENTRAL LIMIT BOUND FOR LINEAR COMBINATIONS OF ORDER STATISTICS**

By

TIEE-JIAN WU, *University of Houston *and* Academia Sinica*

SUMMARY. Under the assumption that the absolute third moment of the underlying distribution is finite, the ideal Lp, $1 \leq p \leq \infty$, nonuniform (with multiplying factor (1 + |x|)^{3-1/p}) central limit bound of order O(N^{-1/2}) is established for linear combinations of order statistics with weight functions being Lipschitz of order 1 on (0,1). This extends and refines Theorems 3.1.1-3.1.2 of Helmers (1984).

*AMS (1980) subject classification.* Primary 62G30, 60F05, secondary 62E20.

*Key words and phrases.* $L_p$ nonuniform central limit bounds, Linear combinations of order statistics.