  This paper develops a general framework for analyzing asymptotics of
$V$-statistics. Previous literature on limiting distribution mainly focuses on
the cases when $n \to \infty$ with fixed kernel size $k$. Under some regularity
conditions, we demonstrate asymptotic normality when $k$ grows with $n$ by
utilizing existing results for $U$-statistics. The key in our approach lies in
a mathematical reduction to $U$-statistics by designing an equivalent kernel
for $V$-statistics. We also provide a unified treatment on variance estimation
for both $U$- and $V$-statistics by observing connections to existing methods
and proposing an empirically more accurate estimator. Ensemble methods such as
random forests, where multiple base learners are trained and aggregated for
prediction purposes, serve as a running example throughout the paper because
they are a natural and flexible application of $V$-statistics.
