  We analyze the fluctuations of incomplete $U$-statistics over a triangular
array of independent random variables. We give criteria for a Central Limit
Theorem (CLT, for short) to hold in the sense that we prove that an
appropriately scaled and centered version of the U-statistic converges to a
normal random variable. Our method of proof relies on a martingale CLT. A
possible application -- a CLT for the hitting time for random walk on random
graphs -- will be presented in \cite{LoTe20b}
