  In this paper, weak convergences of marked empirical processes in
$L^2(\mathbb{R},\nu)$ and their applications to statistical goodness-of-fit
tests are provided, where $L^2(\mathbb{R},\nu)$ is the set of equivalence
classes of the square integrable functions on $\mathbb{R}$ with respect to a
finite Borel measure $\nu$. The results obtained in our framework of weak
convergences are, in the topological sense, weaker than those in the Skorokhod
topology on a space of c\'adl\'ag functions or the uniform topology on a space
of bounded functions, which have been well studied in previous works. However,
our results have the following merits: (1) avoiding conditions which do not
suit for our purpose; (2) treating a weight function which makes us possible to
propose an Anderson--Darling type test statistics for goodness-of-fit tests.
Indeed, the applications presented in this paper are novel.
