Let $H(p)$ be the set ${xin X:; h(x)leq p$, where $h$ is a real-valued lower semicontinuous function on a locally compact second countable metric space $X$. A limit theorem is proved for the empirical counterpart of $H(p)$ obtained by replacing of $h$ with its estimator.

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CWI
Department of Operations Research, Statistics, and System Theory [BS]

Molchanov, I.S. (1995). A limit theorem for solutions of inequalities. Department of Operations Research, Statistics, and System Theory [BS]. CWI.