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Conjecture: Lindeberg-type condition suffices for CLT under finite second moments

Prove that a suitable Lindeberg-type condition, together with only a finite second moment assumption on the kernel h, suffices to obtain the central limit theorem for incomplete U-statistics of deterministic designs stated as Theorem \ref{theo:CLT}, thereby removing the need for higher-order moment assumptions.

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Background

The paper’s central limit theorem (Theorem \ref{theo:CLT}) currently requires a (2+ε)-moment bound on the kernel to control normal approximation via dependency graphs and Stein’s method. This condition can be restrictive in applications with heavy-tailed kernels.

Inspired by CLTs for m-dependent processes, the authors conjecture that a Lindeberg-type condition might be sufficient under just finite second moments. Establishing this would broaden applicability and simplify assumptions for asymptotic normality of incomplete U-statistics with deterministic designs.

References

Moreover, inspired by the work of on $m$-dependent processes, we conjecture that a Lindeberg-type condition may suffice to obtain the conclusions of Theorem~\ref{theo:CLT} under only a finite second moment assumption on the kernel.

— Incomplete U-Statistics of Equireplicate Designs: Berry-Esseen Bound and Efficient Construction (2510.20755 - Miglioli et al., 23 Oct 2025) in Section 6 (Conclusion)