The purpose of this paper is to investigate the asymptotic normality of linear combinations of order statistics; that is, to find conditions under which a statistic of the form $S_n = \mathbf{\sum}^n_{i=1} c_{in}X_{in}$ has a limiting normal distribution as $n$ becomes infinite, where the $c_{in}$'s are constants and $X_{1n}, X_{2n}, \cdots, X_{nn}$ are the observations of a sample of size $n$, ordered by increasing magnitude. Aside from the sample mean (the case where the weights $c_{in}$ are all equal to $1/n$), the first proof of asymptotic normality within this class was by Smirnov in 1935 [19], who considered the case that nonzero weight is attached to at most two percentiles. In 1946, Mosteller [13] extended this to the case of several percentiles, and coined the phrase "systematic statistic" to describe $S_n$. Since the publication in 1955 of a paper by Jung [11] concerned with finding optimal weights for $S_n$ in certain estimation problems, interest in proving its asymptotic normality under more general conditions has grown. For example, Weiss in [21] proved that $S_n$ has a limiting normal distribution when no weight is attached to the observations below the $p$th sample percentile and above the $q$th sample percentile, $p < q$, and the remaining observations are weighted according to a function $J$ by $c_{in} = J(i/(n + 1))$, where $J$ is assumed to have a bounded derivative between $p$ and $q$. Within the past few years, several notable attempts have been made to prove the asymptotic normality of $S_n$ under more general conditions on the weights and underlying distribution. These attempts have employed three essentially different methods. In [1] Bickel used an invariance principle for order statistics to prove asymptotic normality when $\mathbf{\sum}_{i