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Large ball probabilities, Gaussian comparison and anti-concentration

Author

Listed:
  • Götze, Friedrich
  • Naumov, Alexey
  • Spokoiny, Vladimir
  • Ulyanov, Vladimir

Abstract

We derive tight non-asymptotic bounds for the Kolmogorov distance between the probabilities of two Gaussian elements to hit a ball in a Hilbert space. The key property of these bounds is that they are dimension-free and depend on the nuclear (Schatten-one) norm of the difference between the covariance operators of the elements and on the norm of the mean shift. The obtained bounds significantly improve the bound based on Pinsker's inequality via the Kullback-Leibler divergence. We also establish an anti-concentration bound for a squared norm of a non-centered Gaussian element in Hilbert space. The paper presents a number of examples motivating our results and applications of the obtained bounds to statistical inference and to high-dimensional CLT.

Suggested Citation

  • Götze, Friedrich & Naumov, Alexey & Spokoiny, Vladimir & Ulyanov, Vladimir, 2018. "Large ball probabilities, Gaussian comparison and anti-concentration," IRTG 1792 Discussion Papers 2018-026, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    • Handle: RePEc:zbw:irtgdp:2018026
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    Cited by:

    1. Qingliang Fan & Wei Zhong, 2018. "Nonparametric Additive Instrumental Variable Estimator: A Group Shrinkage Estimation Perspective," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 36(3), pages 388-399, July.
    2. Zhong, Wei & Liu, Xi & Ma, Shuangge, 2018. "Variable selection and direction estimation for single-index models via DC-TGDR method," IRTG 1792 Discussion Papers 2018-050, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    3. Guo, Li & Tao, Yubo & Härdle, Wolfgang Karl, 2018. "Understanding Latent Group Structure of Cryptocurrencies Market: A Dynamic Network Perspective," IRTG 1792 Discussion Papers 2018-032, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    4. Packham, Natalie & Woebbeking, Fabian, 2018. "A factor-model approach for correlation scenarios and correlation stress-testing," IRTG 1792 Discussion Papers 2018-034, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    5. Xiaojia Bao & Qingliang Fan, 2020. "The impact of temperature on gaming productivity: evidence from online games," Empirical Economics, Springer, vol. 58(2), pages 835-867, February.
    6. Packham, Natalie & Kalkbrener, Michael & Overbeck, Ludger, 2018. "Default probabilities and default correlations under stress," IRTG 1792 Discussion Papers 2018-037, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    7. Kuczmaszewska, Anna & Yan, Ji Gao, 2018. "On complete convergence in Marcinkiewicz-Zygmund type SLLN for random variables," IRTG 1792 Discussion Papers 2018-041, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    8. Chen, Haiqiang & Li, Yingxing & Lin, Ming & Zhu, Yanli, 2018. "A Regime Shift Model with Nonparametric Switching Mechanism," IRTG 1792 Discussion Papers 2018-048, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    9. Yatracos, Yannis G., 2018. "Residual'S Influence Index (Rinfin), Bad Leverage And Unmasking In High Dimensional L2-Regression," IRTG 1792 Discussion Papers 2018-060, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    10. Zbonakova, Lenka & Li, Xinjue & Härdle, Wolfgang Karl, 2018. "Penalized Adaptive Forecasting with Large Information Sets and Structural Changes," IRTG 1792 Discussion Papers 2018-039, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    11. Packham, Natalie, 2018. "Optimal contracts under competition when uncertainty from adverse selection and moral hazard are present," IRTG 1792 Discussion Papers 2018-033, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    12. Cai, Zongwu & Fang, Ying & Lin, Ming & Su, Jia, 2018. "Inferences for a Partially Varying Coefficient Model With Endogenous Regressors," IRTG 1792 Discussion Papers 2018-047, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    13. Wang, Honglin & Yu, Fan & Zhou, Yinggang, 2018. "Property Investment and Rental Rate under Housing Price Uncertainty: A Real Options Approach," IRTG 1792 Discussion Papers 2018-051, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    14. Yan, Ji Gao, 2018. "Complete Convergence and Complete Moment Convergence for Maximal Weighted Sums of Extended Negatively Dependent Random Variables," IRTG 1792 Discussion Papers 2018-040, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    15. Kalkbrener, Michael & Packham, Natalie, 2018. "Correlation Under Stress In Normal Variance Mixture Models," IRTG 1792 Discussion Papers 2018-035, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    16. Chiu, Hsin-Yu & Chiang, Mi-Hsiu & Kuo, Wei-Yu, 2018. "Predicative Ability of Similarity-based Futures Trading Strategies," IRTG 1792 Discussion Papers 2018-045, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    17. Guo, Shaojun & Li, Dong & Li, Muyi, 2018. "Strict Stationarity Testing and GLAD Estimation of Double Autoregressive Models," IRTG 1792 Discussion Papers 2018-049, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    18. Koziuk, Andzhey & Spokoiny, Vladimir, 2018. "Toolbox: Gaussian comparison on Eucledian balls," IRTG 1792 Discussion Papers 2018-028, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".

    More about this item

    Keywords

    Gaussian comparison; Gaussian anti-concentration inequalities; effective rank; dimension free bounds; Schatten norm; high-dimensional inference;
    All these keywords.

    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General

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