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Yurinskii's Coupling for Martingales

Author

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  • Matias D. Cattaneo
  • Ricardo P. Masini
  • William G. Underwood

Abstract

Yurinskii's coupling is a popular theoretical tool for non-asymptotic distributional analysis in mathematical statistics and applied probability, offering a Gaussian strong approximation with an explicit error bound under easily verifiable conditions. Originally stated in $\ell^2$-norm for sums of independent random vectors, it has recently been extended both to the $\ell^p$-norm, for $1 \leq p \leq \infty$, and to vector-valued martingales in $\ell^2$-norm, under some strong conditions. We present as our main result a Yurinskii coupling for approximate martingales in $\ell^p$-norm, under substantially weaker conditions than those previously imposed. Our formulation further allows for the coupling variable to follow a more general Gaussian mixture distribution, and we provide a novel third-order coupling method which gives tighter approximations in certain settings. We specialize our main result to mixingales, martingales, and independent data, and derive uniform Gaussian mixture strong approximations for martingale empirical processes. Applications to nonparametric partitioning-based and local polynomial regression procedures are provided, alongside central limit theorems for high-dimensional martingale vectors.

Suggested Citation

  • Matias D. Cattaneo & Ricardo P. Masini & William G. Underwood, 2022. "Yurinskii's Coupling for Martingales," Papers 2210.00362, arXiv.org, revised Sep 2024.
  • Handle: RePEc:arx:papers:2210.00362
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    References listed on IDEAS

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    3. Victor Chernozhukov & Denis Chetverikov & Kengo Kato & Yuta Koike, 2019. "Improved Central Limit Theorem and bootstrap approximations in high dimensions," Papers 1912.10529, arXiv.org, revised May 2022.
    4. Belloni, Alexandre & Chernozhukov, Victor & Chetverikov, Denis & Fernández-Val, Iván, 2019. "Conditional quantile processes based on series or many regressors," Journal of Econometrics, Elsevier, vol. 213(1), pages 4-29.
    5. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors," Papers 1212.6906, arXiv.org, revised Jan 2018.
    6. Li, Jia & Liao, Zhipeng, 2020. "Uniform nonparametric inference for time series," Journal of Econometrics, Elsevier, vol. 219(1), pages 38-51.
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    Cited by:

    1. Demian Pouzo, 2024. "Maximal Inequalities for Empirical Processes under General Mixing Conditions with an Application to Strong Approximations," Papers 2402.11394, arXiv.org, revised Apr 2024.
    2. Giovanni Ballarin, 2023. "Impulse Response Analysis of Structural Nonlinear Time Series Models," Papers 2305.19089, arXiv.org, revised Jun 2024.
    3. Luis Alvarez & Cristine Pinto, 2023. "A maximal inequality for local empirical processes under weak dependence," Papers 2307.01328, arXiv.org.

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