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A Berry–Esseen bound for vector-valued martingales

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  • Kojevnikov, Denis
  • Song, Kyungchul

Abstract

This note provides a conditional Berry–Esseen bound for the sum of a martingale difference sequence {Xi}i=1n in Rd, d≥1, adapted to a filtration {Fi}i=1n. We approximate the conditional distribution of S=∑i=1nXi given a sub-σ-field F0⊂F1 by that of a mean zero normal random vector having the same conditional variance given F0 as the vector S. Assuming that the conditional variances E[XiXi⊤∣Fi−1], i≥1, are F0-measurable and non-singular, and the third conditional moments of ‖Xi‖, i≥1, given F0 are uniformly bounded, we present a simple bound on the conditional Kolmogorov distance between S and its approximation given F0 which is of order Oa.s.([ln(ed)]5/4n−1/4).

Suggested Citation

  • Kojevnikov, Denis & Song, Kyungchul, 2022. "A Berry–Esseen bound for vector-valued martingales," Statistics & Probability Letters, Elsevier, vol. 186(C).
    • Handle: RePEc:eee:stapro:v:186:y:2022:i:c:s0167715222000517
    DOI: 10.1016/j.spl.2022.109448
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    References listed on IDEAS

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    1. 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.
    2. 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.
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