Bootstrap validity for CCAR

Establish bootstrap consistency for continuous covariate-adaptive randomization by obtaining the required conditional control of the invariant distribution $\pi_{\Lambda}$, thereby determining whether the bootstrap variance adjustment is valid for CCAR.

Background

The paper proves bootstrap consistency for complete randomization and covered discrete covariate-adaptive randomization procedures. For CCAR, the authors report empirical success but do not prove validity because the argument requires control of the invariant distribution of the imbalance Markov chain under bootstrap resampling.

References

Bootstrap consistency for CCAR works empirically in Section \ref{sec:numerical}. Although it is conjectured that bootstrap can be applied to CCAR \citep[Remark 4.4]{ma2024new}, the extension requires conditional control of the invariant-distribution $\pi_{\Lambda}$.

— Discretization in covariate-adaptive randomization: gains and losses  (2609.11012 - Zhao et al., 10 Sep 2026) in Section 5, remark following Theorem 5.1; see also Appendix, Section “Bootstrap consistency”