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

We study (asymmetric) $U$-statistics based on a stationary sequence of\n$m$-dependent variables; moreover, we consider constrained $U$-statistics,\nwhere the defining multiple sum only includes terms satisfying some\nrestrictions on the gaps between indices. Results include a law of large\nnumbers and a central limit theorem. Special attention is paid to degenerate\ncases where, after the standard normalization, the asymptotic variance\nvanishes; in these cases non-normal limits occur after a different\nnormalization.\n The results are motivated by applications to pattern matching in random\nstrings and permutations. We obtain both new results and new proofs of old\nresults.\n