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

TL;DR

This paper investigates the asymptotic behavior of $U$-statistics for $m$-dependent and constrained sequences, providing limit theorems and applications to pattern matching in random strings and permutations.

Contribution

It introduces new limit theorems for constrained and $m$-dependent $U$-statistics, including degenerate cases with non-normal limits, with applications to pattern matching.

Findings

Law of large numbers for constrained $U$-statistics

Central limit theorem for $m$-dependent sequences

Identification of non-normal limits in degenerate cases

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. 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.