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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

Part of: Theory of computing Combinatorics Combinatorial probability Limit theorems

Published online by Cambridge University Press:  28 March 2023

Svante Janson Open the ORCID record for Svante Janson [Opens in a new window]
Svante Janson*
Affiliation:
Uppsala University
*
*Postal address: Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden. Email address: svante.janson@math.uu.se
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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.

Keywords

Random wordsconstrained occurrencesvincular patterns

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 05A05: Permutations, words, matrices 60C05: Combinatorial probability 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 3 , September 2023 , pp. 841 - 894
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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