  Under left truncation, data $(X_i,Y_i)$ are observed only when $Y_i\le X_i$.
Usually, the distribution function $F$ of the $X_i$ is the target of interest.
In this paper, we study linear functionals $\int\varphi \mathrm{d}F_n$ of the
nonparametric maximum likelihood estimator (MLE) of $F$, the Lynden-Bell
estimator $F_n$. A useful representation of $\int \varphi \mathrm{d}F_n$ is
derived which yields asymptotic normality under optimal moment conditions on
the score function $\varphi$. No continuity assumption on $F$ is required. As a
by-product, we obtain the distributional convergence of the Lynden-Bell
empirical process on the whole real line.
