1 results
Asymptotic normality for
$\boldsymbol{m}$-dependent and constrained 
$\boldsymbol{U}$-statistics, with applications to pattern matching in random strings and permutations
- Part of
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- Journal:
- Advances in Applied Probability / Volume 55 / Issue 3 / September 2023
- Published online by Cambridge University Press:
- 28 March 2023, pp. 841-894
- Print publication:
- September 2023
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- Article
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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.
