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Asymptotic normality for $\boldsymbol{m}$-dependent and constrained $\boldsymbol{U}$-statistics, with applications to pattern matching in random strings and permutations
Published online by Cambridge University Press: 28 March 2023
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.
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust
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