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.

Random convex sets and integral geometry (msc 52A22), Geometric probability and stochastic geometry (msc 60D05), Central limit and other weak theorems (msc 60F05), None of the above, but in MSC2010 section 62Gxx (msc 62G99)
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.