On the asymptotic normality of finite population <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$L$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>L</mi> </math> </EquationSource> </InlineEquation>-statistics

We give sufficient conditions for the asymptotic normality of linear combinations of order statistics (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$L$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>L</mi> </math> </EquationSource> </InlineEquation>-statistics) in the case of simple random samples without replacement. In the first case, restrictions are imposed on the weights of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$L$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>L</mi> </math> </EquationSource> </InlineEquation>-statistics. The second case is on trimmed means, where we introduce a new finite population smoothness condition. Copyright Springer-Verlag Berlin Heidelberg 2014