Asymptotic normality for $\boldsymbol{m}$ -dependent and constrained $\boldsymbol{U}$ -statistics, with applications to pattern matching in random strings and permutations

Abstract We study (asymmetric) $U$ -statistics based on a stationary sequence of $m$ -dependent variables; moreover, we consider constrained $U$ -statistics, where the defining multiple sum only includes terms satisfying some restrictions on the gaps between indices. Results include a law of large numbers and a central limit theorem, together with results on rate of convergence, moment convergence, functional convergence, and a renewal theory version. Special attention is paid to degenerate cases where, after the standard normalization, the asymptotic variance vanishes; in these cases non-normal limits occur after a different normalization. The results are motivated by applications to pattern matching in random strings and permutations. We obtain both new results and new proofs of old results.