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arxiv: 2606.08498 · v1 · pith:6OEERSOTnew · submitted 2026-06-07 · 🧮 math.ST · stat.ME· stat.TH

Tests for Independence of High-Dimensional Nonstationary Time Series

Yunyi Zhang This is my paper

Pith reviewed 2026-06-27 18:03 UTC · model grok-4.3

classification 🧮 math.ST stat.MEstat.TH
keywords independence testinghigh-dimensional time seriesnonstationary processesweighted average statisticdependent wild bootstrapconcentration inequalityquadratic forms
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The pith

A bimodal weighted-average test statistic tests independence between high-dimensional time series without assuming stationarity or whitening the data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a test for whether two high-dimensional time series are independent when their autocovariances are allowed to change over time. The central device is a bimodal weighted-average statistic that cancels the bias coming from temporal dependence under the null, so the usual whitening step can be skipped. The approach is needed because whitening becomes unreliable once dimension is large and stationarity fails, situations that arise in many observed series. Theory supplies a concentration inequality for quadratic forms under a class of nonlinear nonstationary processes and uses it to justify both the asymptotic null distribution and a dependent wild bootstrap. Simulations confirm that size stays correct and power is good even when the number of variables exceeds the number of observations or autocovariances vary.

Core claim

The manuscript claims that a bimodal weighted-average test statistic removes the bias induced by temporal dependence under the null hypothesis, allowing valid inference for independence via a dependent wild bootstrap whose validity follows from a derived concentration inequality for quadratic forms of high-dimensional nonlinear nonstationary time series, with the result that the test attains correct size and power even when dimension exceeds sample size or autocovariances vary over time.

What carries the argument

The bimodal weighted-average test statistic that removes bias induced by temporal dependence under the null hypothesis.

If this is right

  • The test maintains correct size and good power when the dimension exceeds the sample size.
  • The test continues to work when the data-generating process has time-varying autocovariances.
  • Tests based on whitening the series fail to keep correct size once autocovariance structures become unstable.
  • The dependent wild bootstrap remains valid under the concentration inequality established for the quadratic forms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same concentration inequality could support other quadratic-form-based procedures for nonstationary high-dimensional data.
  • Analysts dealing with economic or environmental series that exhibit changing dependence patterns could apply the test without first imposing stationarity.

Load-bearing premise

The time series must belong to the class of high-dimensional nonlinear nonstationary processes for which a concentration inequality for quadratic forms can be derived.

What would settle it

A simulation study in which the empirical rejection rate of the test under the null deviates substantially from the nominal level for processes whose time-varying autocovariances fall outside the conditions required by the concentration inequality.

Figures

Figures reproduced from arXiv: 2606.08498 by Yunyi Zhang.

Figure 1
Figure 1. Figure 1: plots the dynamics of the bias of the test statistic under both 𝐻0 and 𝐻1 , in agreement with Remarks 1 and 2. Under the null, the bias is driven by the autocovariances, and increasing 𝜆1 renders this bias negligible. In contrast, under 𝐻1 , the presence of nonzero cross-covariance matrices induces extra bias in the test statistic, thereby inflating its absolute value. (a) log(Bias) under 𝐻0 . (b) log(Bias… view at source ↗
Figure 2
Figure 2. Figure 2: Population and estimated causal graphs for the data-generating process ( [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Observation examples and hypothesis testing results on S&P 500 stock prices. [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
read the original abstract

This manuscript studies the problem of independence testing between two high-dimensional time series without assuming weak stationarity, that is, allowing their autocovariances to vary over time. To this end, we propose a bimodal weighted-average test statistic that removes the bias induced by temporal dependence under the null hypothesis, thereby avoiding the need to whiten the time series prior to hypothesis testing -- a procedure that is challenging in high-dimensional and nonstationary settings. To facilitate statistical inference, we develop a dependent wild bootstrap procedure. On the theoretical side, we derive a concentration inequality for quadratic forms of time series data stemming from a class of high-dimensional, nonlinear, and nonstationary processes. This result enables us to derive the asymptotic null distribution of the proposed test statistic and to establish the validity of the bootstrap algorithm. Numerical results show that the proposed test attains desired size and good power performance even when the dimension exceeds the sample size or when the data-generating process exhibits time-varying autocovariances. In contrast, tests based on whitening time series fail to maintain correct size in the presence of unstable autocovariance structures. Since nonstationary autocovariances commonly arise in real-life time series data, our work offers a robust procedure for independence testing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a test for independence between two high-dimensional time series without assuming weak stationarity. It proposes a bimodal weighted-average test statistic that corrects for bias induced by temporal dependence under the null, thereby avoiding explicit whitening. A dependent wild bootstrap is introduced whose validity is established using a derived concentration inequality for quadratic forms of high-dimensional, nonlinear, nonstationary processes. This yields the asymptotic null distribution of the statistic. Simulations indicate that the test maintains correct size and exhibits good power when the dimension exceeds the sample size or when autocovariances vary over time, in contrast to whitening-based competitors.

Significance. If the concentration inequality and bootstrap validity hold under the stated process class, the work supplies a practical procedure for a common setting in which nonstationary autocovariances appear and high dimensionality renders whitening infeasible. The concentration inequality itself is a technical contribution that supports both the asymptotic theory and the bootstrap. The simulation evidence directly addresses the regimes where existing methods break down.

minor comments (2)
  1. The precise definition of the bimodal weighted-average statistic and the choice of weights should be stated with explicit formulas early in the main text rather than deferred.
  2. Notation for the time-varying autocovariance functions and the dependence measures appearing in the concentration inequality should be introduced consistently between the theoretical sections and the simulation design.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so we have no individual points requiring rebuttal or revision at this stage. We are pleased that the contribution of the concentration inequality and the practical advantages in high-dimensional nonstationary settings are recognized.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript derives a concentration inequality for quadratic forms of high-dimensional nonlinear nonstationary processes, then uses it to obtain the asymptotic null distribution of a bias-corrected bimodal weighted-average statistic and to justify a dependent wild bootstrap. No equation or claim in the abstract reduces the test statistic, its limiting distribution, or the bootstrap validity to a fitted parameter or self-citation by construction. The numerical size and power results are presented as separate empirical checks rather than tautological outputs of the same inputs. The argument chain is therefore self-contained and does not match any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, no invented entities, and only one domain assumption visible at the level of the summary.

axioms (1)
  • domain assumption The processes belong to a class of high-dimensional, nonlinear, and nonstationary processes permitting derivation of a concentration inequality for quadratic forms.
    Invoked to obtain the asymptotic null distribution and bootstrap validity (abstract, theoretical side).

pith-pipeline@v0.9.1-grok · 5743 in / 1266 out tokens · 17930 ms · 2026-06-27T18:03:09.972660+00:00 · methodology

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Works this paper leans on

219 extracted references · 167 canonical work pages

  1. [1]

    Berrendero, J. R. and Cuevas, A. and Torrecilla, J L. , title =

  2. [2]

    2025 , note =

    R: A Language and Environment for Statistical Computing , author =. 2025 , note =

    2025

  3. [3]

    D. R. Cox , Journal =

  4. [4]

    and Holmes, Christopher C

    Heard, Nicholas A. and Holmes, Christopher C. and Stephens, David A. , Journal =. A Quantitative Study of Gene Regulation Involved in the Immune Response of

  5. [5]

    Biometrika , FJOURNAL =

    Hong, Yongmiao , TITLE =. Biometrika , FJOURNAL =. 1996 , NUMBER =. doi:10.1093/biomet/83.3.615 , URL =

    work page doi:10.1093/biomet/83.3.615 1996

  6. [6]

    Kulperger, R. J. and Lockhart, R. A. , TITLE =. J. Time Ser. Anal. , FJOURNAL =. 1998 , NUMBER =. doi:10.1111/1467-9892.00084 , URL =

    work page doi:10.1111/1467-9892.00084 1998

  7. [7]

    Duchesne, Pierre and Roy, Roch , TITLE =. Statist. Sinica , FJOURNAL =. 2003 , NUMBER =

    2003

  8. [8]

    Brock, W. A. and Scheinkman, J. A. and Dechert, W. D. and LeBaron, B. , TITLE =. Econometric Rev. , FJOURNAL =. 1996 , NUMBER =. doi:10.1080/07474939608800353 , URL =

    work page doi:10.1080/07474939608800353 1996

  9. [9]

    Econometric Theory , FJOURNAL =

    Shao, Xiaofeng , TITLE =. Econometric Theory , FJOURNAL =. 2009 , NUMBER =. doi:10.1017/S0266466608090063 , URL =

    work page doi:10.1017/s0266466608090063 2009

  10. [10]

    and Yang, Shie-Shien , TITLE =

    Koch, Paul D. and Yang, Shie-Shien , TITLE =. J. Amer. Statist. Assoc. , FJOURNAL =. 1986 , NUMBER =

    1986

  11. [11]

    Bouhaddioui, Chafik and Roy, Roch , TITLE =. J. Time Ser. Anal. , FJOURNAL =. 2006 , NUMBER =. doi:10.1111/j.1467-9892.2006.00473.x , URL =

    work page doi:10.1111/j.1467-9892.2006.00473.x 2006

  12. [12]

    and Fisher, Thomas J

    Robbins, Michael W. and Fisher, Thomas J. , TITLE =. J. Bus. Econom. Statist. , FJOURNAL =. 2015 , NUMBER =. doi:10.1080/07350015.2014.962699 , URL =

    work page doi:10.1080/07350015.2014.962699 2015

  13. [13]

    PLOS Biology , publisher =

    A rigorous and versatile statistical test for correlations between stationary time series , year =. PLOS Biology , publisher =. doi:10.1371/journal.pbio.3002758 , author =

    work page doi:10.1371/journal.pbio.3002758

  14. [14]

    A brain-wide map of neural activity during complex behaviour , url =

    Meshulam, Leenoy and Angelaki, Dora and Benson, Julius and McRoberts, Isaiah and Noel, Jean-Paul and Arlandis, Jaime and Bonacchi, Niccol. A brain-wide map of neural activity during complex behaviour , url =. Nature , number =. 2025 , bdsk-url-1 =. doi:10.1038/s41586-025-09235-0 , id =

    work page doi:10.1038/s41586-025-09235-0 2025

  15. [15]

    and Hu, Sarah K

    Coenen, Ashley R. and Hu, Sarah K. and Luo, Elaine and Muratore, Daniel and Weitz, Joshua S. , journal =. A Primer for Microbiome Time-Series Analysis , year =

  16. [16]

    and Brock, Melissa L

    Larkin, Alyse A. and Brock, Melissa L. and Fagan, Adam J. and Moreno, Allison R. and Gerace, Skylar D. and Lees, Lauren E. and Suarez, Stacy A. and Eloe-Fadrosh, Emiley A. and Martiny, Adam C. , date =. Climate-driven succession in marine microbiome biodiversity and biogeochemical function , url =. Nature Communications , number =. 2025 , bdsk-url-1 =. do...

    work page doi:10.1038/s41467-025-59382-1 2025

  17. [17]

    Wang, Guochang and Li, Wai Keung and Zhu, Ke , TITLE =. Statist. Sinica , FJOURNAL =. 2021 , NUMBER =. doi:10.5705/ss.20 , URL =

    work page doi:10.5705/ss.20 2021

  18. [18]

    Chu, Ba , TITLE =. J. Multivariate Anal. , FJOURNAL =. 2023 , PAGES =. doi:10.1016/j.jmva.2022.105151 , URL =

    work page doi:10.1016/j.jmva.2022.105151 2023

  19. [19]

    Meintanis, Simos G. and Hu. Fourier-type tests of mutual independence between functional time series , JOURNAL =. 2022 , PAGES =. doi:10.1016/j.jmva.2021.104873 , URL =

    work page doi:10.1016/j.jmva.2021.104873 2022

  20. [20]

    2003 , PAGES =

    Shao, Jun , TITLE =. 2003 , PAGES =. doi:10.1007/b97553 , URL =

    work page doi:10.1007/b97553 2003

  21. [21]

    and Johnson, Charles R

    Horn, Roger A. and Johnson, Charles R. , TITLE =. 2013 , PAGES =

    2013

  22. [22]

    , TITLE =

    Zhang, Cun-Hui and Zhang, Stephanie S. , TITLE =. J. R. Stat. Soc. Ser. B. Stat. Methodol. , FJOURNAL =. 2014 , NUMBER =. doi:10.1111/rssb.12026 , URL =

    work page doi:10.1111/rssb.12026 2014

  23. [24]

    Krampe, Jonas and Paparoditis, Efstathios , TITLE =. J. Time Series Anal. , FJOURNAL =. 2021 , NUMBER =. doi:10.1111/jtsa.12586 , URL =

    work page doi:10.1111/jtsa.12586 2021

  24. [25]

    Basu, Sumanta and Michailidis, George , TITLE =. Ann. Statist. , FJOURNAL =. 2015 , NUMBER =. doi:10.1214/15-AOS1315 , URL =

    work page doi:10.1214/15-aos1315 2015

  25. [26]

    2003 , PAGES =

    ksendal, Bernt , TITLE =. 2003 , PAGES =. doi:10.1007/978-3-642-14394-6 , URL =

    work page doi:10.1007/978-3-642-14394-6 2003

  26. [27]

    , TITLE =

    Whittle, P. , TITLE =. Teor. Verojatnost. i Primenen. , FJOURNAL =. 1960 , PAGES =

    1960

  27. [28]

    Burkholder, D. L. and Davis, B. J. and Gundy, R. F. , TITLE =. Proceedings of the. 1972 , MRCLASS =

    1972

  28. [29]

    Biometrika , FJOURNAL =

    Xu, Mengyu and Zhang, Danna and Wu, Wei Biao , TITLE =. Biometrika , FJOURNAL =. 2019 , NUMBER =. doi:10.1093/biomet/asz020 , URL =

    work page doi:10.1093/biomet/asz020 2019

  29. [30]

    Chernozhukov, Victor and Chetverikov, Denis and Kato, Kengo , TITLE =. Ann. Statist. , FJOURNAL =. 2014 , NUMBER =. doi:10.1214/14-AOS1230 , URL =

    work page doi:10.1214/14-aos1230 2014

  30. [31]

    , TITLE =

    Zhang, Yunyi and Politis, Dimitris N. , TITLE =. Ann. Statist. , FJOURNAL =. 2022 , NUMBER =. doi:10.1214/21-aos2156 , URL =

    work page doi:10.1214/21-aos2156 2022

  31. [32]

    Chernozhukov, Victor and Chetverikov, Denis and Kato, Kengo , TITLE =. Probab. Theory Related Fields , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00440-014-0565-9 , URL =

    work page doi:10.1007/s00440-014-0565-9 2015

  32. [33]

    , TITLE =

    Folland, Gerald B. , TITLE =. 1999 , PAGES =

    1999

  33. [34]

    Zhang, Danna and Wu, Wei Biao , TITLE =. Ann. Statist. , FJOURNAL =. 2021 , NUMBER =. doi:10.1214/20-AOS1954 , URL =

    work page doi:10.1214/20-aos1954 2021

  34. [35]

    Messner and Pierre Pinson , keywords =

    Jakob W. Messner and Pierre Pinson , keywords =. Online adaptive lasso estimation in vector autoregressive models for high dimensional wind power forecasting , journal =. 2019 , issn =. doi:https://doi.org/10.1016/j.ijforecast.2018.02.001 , url =

    work page doi:10.1016/j.ijforecast.2018.02.001 2019

  35. [36]

    Hannaford and Sarah E

    Naomi E. Hannaford and Sarah E. Heaps and Tom M.W. Nye and Thomas P. Curtis and Ben Allen and Andrew Golightly and Darren J. Wilkinson , keywords =. A sparse Bayesian hierarchical vector autoregressive model for microbial dynamics in a wastewater treatment plant , journal =. 2023 , issn =. doi:https://doi.org/10.1016/j.csda.2022.107659 , url =

    work page doi:10.1016/j.csda.2022.107659 2023

  36. [37]

    Comparing long monthly Chinese and selected European temperature series using the Vector Seasonal Shifting Mean and Covariance Autoregressive model , journal =

    Changli He and Jian Kang and Timo Teräsvirta and Shuhua Zhang , keywords =. Comparing long monthly Chinese and selected European temperature series using the Vector Seasonal Shifting Mean and Covariance Autoregressive model , journal =. 2021 , issn =. doi:https://doi.org/10.1016/j.eneco.2021.105171 , url =

    work page doi:10.1016/j.eneco.2021.105171 2021

  37. [38]

    Exchange Rate: Evidence from a Vector Autoregression , author =

    The Canadian-U.S. Exchange Rate: Evidence from a Vector Autoregression , author =. 1986 , journal =

    1986

  38. [39]

    Campbell and Robert J

    John Y. Campbell and Robert J. Shiller , journal =. Stock Prices, Earnings, and Expected Dividends , urldate =

  39. [40]

    Industrial and Corporate Change , volume =

    Coad, Alex , title = ". Industrial and Corporate Change , volume =. 2010 , month =. doi:10.1093/icc/dtq018 , url =

    work page doi:10.1093/icc/dtq018 2010

  40. [41]

    and Sioshansi Ramteen , title = "

    Liu, Yixian and Roberts, Matthew C. and Sioshansi Ramteen , title = ". J. Mod. Power Syst. Clean Energy , volume =. 2018 , month =

    2018

  41. [43]

    Wang, Di and Zheng, Yao and Lian, Heng and Li, Guodong , TITLE =. J. Amer. Statist. Assoc. , FJOURNAL =. 2022 , NUMBER =. doi:10.1080/01621459.2020.1855183 , URL =

    work page doi:10.1080/01621459.2020.1855183 2022

  42. [44]

    and Davis, Richard A

    Brockwell, Peter J. and Davis, Richard A. , TITLE =. 1991 , PAGES =. doi:10.1007/978-1-4419-0320-4 , URL =

    work page doi:10.1007/978-1-4419-0320-4 1991

  43. [45]

    and Davis, Richard A

    Brockwell, Peter J. and Davis, Richard A. , TITLE =. 2016 , PAGES =. doi:10.1007/978-3-319-29854-2 , URL =

    work page doi:10.1007/978-3-319-29854-2 2016

  44. [46]

    , TITLE =

    Reinsel, Gregory C. , TITLE =. 1997 , PAGES =. doi:10.1007/978-1-4612-0679-8 , URL =

    work page doi:10.1007/978-1-4612-0679-8 1997

  45. [47]

    , TITLE =

    Reinsel, Gregory C. , TITLE =. 1993 , PAGES =. doi:10.1007/978-1-4684-0198-1 , URL =

    work page doi:10.1007/978-1-4684-0198-1 1993

  46. [48]

    Xiao, Han and Wu, Wei Biao , TITLE =. Statist. Sinica , FJOURNAL =. 2014 , NUMBER =

    2014

  47. [49]

    Carlos Escanciano and Ignacio N

    J. Carlos Escanciano and Ignacio N. Lobato , keywords =. An automatic Portmanteau test for serial correlation , journal =. 2009 , note =. doi:https://doi.org/10.1016/j.jeconom.2009.03.001 , url =

    work page doi:10.1016/j.jeconom.2009.03.001 2009

  48. [50]

    Zhao, Peng and Yu, Bin , TITLE =. J. Mach. Learn. Res. , FJOURNAL =. 2006 , PAGES =

    2006

  49. [51]

    Tibshirani, Robert , TITLE =. J. Roy. Statist. Soc. Ser. B , FJOURNAL =. 1996 , NUMBER =

    1996

  50. [52]

    and Medeiros, Marcelo C

    Masini, Ricardo P. and Medeiros, Marcelo C. and Mendes, Eduardo F. , TITLE =. J. Time Series Anal. , FJOURNAL =. 2022 , NUMBER =. doi:10.1111/jtsa.12627 , URL =

    work page doi:10.1111/jtsa.12627 2022

  51. [53]

    Wong, Kam Chung and Li, Zifan and Tewari, Ambuj , TITLE =. Ann. Statist. , FJOURNAL =. 2020 , NUMBER =. doi:10.1214/19-AOS1840 , URL =

    work page doi:10.1214/19-aos1840 2020

  52. [54]

    Kock, Anders Bredahl and Callot, Laurent , TITLE =. J. Econometrics , FJOURNAL =. 2015 , NUMBER =. doi:10.1016/j.jeconom.2015.02.013 , URL =

    work page doi:10.1016/j.jeconom.2015.02.013 2015

  53. [55]

    Bernoulli , FJOURNAL =

    Krampe, Jonas and Kreiss, Jens-Peter and Paparoditis, Efstathios , TITLE =. Bernoulli , FJOURNAL =. 2021 , NUMBER =. doi:10.3150/20-bej1239 , URL =

    work page doi:10.3150/20-bej1239 2021

  54. [56]

    Han, Fang and Lu, Huanran and Liu, Han , TITLE =. J. Mach. Learn. Res. , FJOURNAL =. 2015 , PAGES =

    2015

  55. [57]

    Cai, Tony and Liu, Weidong and Luo, Xi , TITLE =. J. Amer. Statist. Assoc. , FJOURNAL =. 2011 , NUMBER =. doi:10.1198/jasa.2011.tm10155 , URL =

    work page doi:10.1198/jasa.2011.tm10155 2011

  56. [58]

    Biometrika , FJOURNAL =

    Guo, Shaojun and Wang, Yazhen and Yao, Qiwei , TITLE =. Biometrika , FJOURNAL =. 2016 , NUMBER =. doi:10.1093/biomet/asw046 , URL =

    work page doi:10.1093/biomet/asw046 2016

  57. [59]

    Biometrika , FJOURNAL =

    Zheng, Yao and Cheng, Guang , TITLE =. Biometrika , FJOURNAL =. 2021 , NUMBER =. doi:10.1093/biomet/asaa065 , URL =

    work page doi:10.1093/biomet/asaa065 2021

  58. [60]

    Lin, Jiahe and Michailidis, George , TITLE =. J. Mach. Learn. Res. , FJOURNAL =. 2017 , PAGES =. doi:10.1631/jzus.a1500279 , URL =

    work page doi:10.1631/jzus.a1500279 2017

  59. [61]

    and Paparoditis, E

    Krampe, J. and Paparoditis, E. and Trenkler, C. , TITLE =. J. Econometrics , FJOURNAL =. 2023 , NUMBER =. doi:10.1016/j.jeconom.2022.01.003 , URL =

    work page doi:10.1016/j.jeconom.2022.01.003 2023

  60. [62]

    Zhu, Xuening and Pan, Rui and Li, Guodong and Liu, Yuewen and Wang, Hansheng , TITLE =. Ann. Statist. , FJOURNAL =. 2017 , NUMBER =. doi:10.1214/16-AOS1476 , URL =

    work page doi:10.1214/16-aos1476 2017

  61. [63]

    and Levina, Elizaveta , TITLE =

    Bickel, Peter J. and Levina, Elizaveta , TITLE =. Ann. Statist. , FJOURNAL =. 2008 , NUMBER =. doi:10.1214/08-AOS600 , URL =

    work page doi:10.1214/08-aos600 2008

  62. [64]

    , TITLE =

    Kreiss, Jens-Peter and Paparoditis, Efstathios and Politis, Dimitris N. , TITLE =. Ann. Statist. , FJOURNAL =. 2011 , NUMBER =. doi:10.1214/11-AOS900 , URL =

    work page doi:10.1214/11-aos900 2011

  63. [65]

    , TITLE =

    Wang, Jiang and Politis, Dimitris N. , TITLE =. J. Time Series Anal. , FJOURNAL =. 2021 , NUMBER =. doi:10.1111/jtsa.12580 , URL =

    work page doi:10.1111/jtsa.12580 2021

  64. [66]

    and Politis, Dimitris N

    McMurry, Timothy L. and Politis, Dimitris N. , TITLE =. J. Time Series Anal. , FJOURNAL =. 2010 , NUMBER =. doi:10.1111/j.1467-9892.2010.00679.x , URL =

    work page doi:10.1111/j.1467-9892.2010.00679.x 2010

  65. [67]

    , TITLE =

    Jentsch, Carsten and Politis, Dimitris N. , TITLE =. Ann. Statist. , FJOURNAL =. 2015 , NUMBER =. doi:10.1214/14-AOS1301 , URL =

    work page doi:10.1214/14-aos1301 2015

  66. [68]

    Fragkeskou, Maria and Paparoditis, Efstathios , TITLE =. J. Time Series Anal. , FJOURNAL =. 2018 , NUMBER =. doi:10.1111/jtsa.12275 , URL =

    work page doi:10.1111/jtsa.12275 2018

  67. [69]

    and Politis, Dimitris N

    McMurry, Timothy L. and Politis, Dimitris N. , TITLE =. Electron. J. Stat. , FJOURNAL =. 2015 , NUMBER =. doi:10.1214/15-EJS1000 , URL =

    work page doi:10.1214/15-ejs1000 2015

  68. [70]

    Zhu, Ke , TITLE =. Ann. Statist. , FJOURNAL =. 2019 , NUMBER =. doi:10.1214/18-AOS1775 , URL =

    work page doi:10.1214/18-aos1775 2019

  69. [71]

    Journal of the American Statistical Association , author =

    Das, Srinjoy and Politis, Dimitris N. , TITLE =. J. Amer. Statist. Assoc. , FJOURNAL =. 2021 , NUMBER =. doi:10.1080/01621459.2019.1708368 , URL =

    work page doi:10.1080/01621459.2019.1708368 2021

  70. [72]

    Kreiss, Jens-Peter and Paparoditis, Efstathios , TITLE =. J. R. Stat. Soc. Ser. B. Stat. Methodol. , FJOURNAL =. 2015 , NUMBER =. doi:10.1111/rssb.12068 , URL =

    work page doi:10.1111/rssb.12068 2015

  71. [73]

    Bernoulli , FJOURNAL =

    Dahlhaus, Rainer and Richter, Stefan and Wu, Wei Biao , TITLE =. Bernoulli , FJOURNAL =. 2019 , NUMBER =. doi:10.3150/17-bej1011 , URL =

    work page doi:10.3150/17-bej1011 2019

  72. [74]

    Lai, Tze Leung and Liu, Haiyan and Xing, Haipeng , TITLE =. Statist. Sinica , FJOURNAL =. 2005 , NUMBER =

    2005

  73. [75]

    Arxiv id: 2112.00693 , year=

    Xiucai Ding and Zhou Zhou , title=. Arxiv id: 2112.00693 , year=

    arXiv

  74. [76]

    Functional weak limit theorem for a local empirical process of non-stationary time series and its application , JOURNAL =

    Mayer, Ulrike and Z\". Functional weak limit theorem for a local empirical process of non-stationary time series and its application , JOURNAL =. 2020 , NUMBER =. doi:10.3150/19-BEJ1174 , URL =

    work page doi:10.3150/19-bej1174 2020

  75. [77]

    Wu, Wei Biao and Zhou, Zhou , TITLE =. Statist. Sinica , FJOURNAL =. 2011 , NUMBER =. doi:10.5705/ss.2008.223 , URL =

    work page doi:10.5705/ss.2008.223 2011

  76. [78]

    Ding, Xiucai and Zhou, Zhou , TITLE =. Ann. Statist. , FJOURNAL =. 2020 , NUMBER =. doi:10.1214/19-AOS1894 , URL =

    work page doi:10.1214/19-aos1894 2020

  77. [79]

    2023 , journal=

    Central limit theorems for high dimensional dependent data , author=. 2023 , journal=

    2023

  78. [80]

    2022 , journal=

    Sequential Gaussian approximation for nonstationary time series in high dimensions , author=. 2022 , journal=

    2022

  79. [81]

    Zudi Lu and Xiaohang Ren and Rongmao Zhang , title =. J. Amer. Statist. Assoc. , volume =. 2023 , publisher =. doi:10.1080/01621459.2022.2161386 , URL =

    work page doi:10.1080/01621459.2022.2161386 2023

  80. [82]

    Electron

    Liu, Hanzhong and Yu, Bin , TITLE =. Electron. J. Stat. , FJOURNAL =. 2013 , PAGES =. doi:10.1214/14-EJS875 , URL =

    work page doi:10.1214/14-ejs875 2013

Showing first 80 references.