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Exponential inequalities for sums of random vectors


  • Yurinskii, V. V.


This paper presents some generalizations of S. N. Bernstein's exponential bounds on probabilities of large deviations to the vector case. Inequalities for probabilities of large deviations of sums of independent random vectors are derived under a Cramér's type restriction on the rate of growth of absolute moments of the summands. Estimates are obtained for random vectors with values in Banach space, Sharper bounds hold in the case of finite-dimensional Euclidean or separable Hilbert spaces.

Suggested Citation

  • Yurinskii, V. V., 1976. "Exponential inequalities for sums of random vectors," Journal of Multivariate Analysis, Elsevier, vol. 6(4), pages 473-499, December.
  • Handle: RePEc:eee:jmvana:v:6:y:1976:i:4:p:473-499

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    References listed on IDEAS

    1. Robinson, P M, 1987. "Asymptotically Efficient Estimation in the Presence of Heteroskedasticity of Unknown Form," Econometrica, Econometric Society, vol. 55(4), pages 875-891, July.
    2. Pollard, David, 1985. "New Ways to Prove Central Limit Theorems," Econometric Theory, Cambridge University Press, vol. 1(03), pages 295-313, December.
    3. Andrews, Donald W. K., 1988. "Chi-square diagnostic tests for econometric models : Introduction and applications," Journal of Econometrics, Elsevier, vol. 37(1), pages 135-156, January.
    4. Andrews, Donald W K, 1988. "Chi-Square Diagnostic Tests for Econometric Models: Theory," Econometrica, Econometric Society, vol. 56(6), pages 1419-1453, November.
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    Cited by:

    1. Chen, Xia, 1997. "Moderate deviations for m-dependent random variables with Banach space values," Statistics & Probability Letters, Elsevier, vol. 35(2), pages 123-134, September.
    2. Otsu, Taisuke, 2011. "Moderate deviations of generalized method of moments and empirical likelihood estimators," Journal of Multivariate Analysis, Elsevier, vol. 102(8), pages 1203-1216, September.
    3. Cardot, Hervé & Sarda, Pacal, 2005. "Estimation in generalized linear models for functional data via penalized likelihood," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 24-41, January.
    4. Boente, Graciela & Fraiman, Ricardo, 2000. "Kernel-based functional principal components," Statistics & Probability Letters, Elsevier, vol. 48(4), pages 335-345, July.
    5. Ahmedou, Aziza & Marion, Jean-Marie & Pumo, Besnik, 2016. "Generalized linear model with functional predictors and their derivatives," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 313-324.
    6. Inglot, Tadeusz, 2000. "On large deviation theorem for data-driven Neyman's statistic," Statistics & Probability Letters, Elsevier, vol. 47(4), pages 411-419, May.
    7. Cardot, Hervé & Ferraty, Frédéric & Sarda, Pascal, 1999. "Functional linear model," Statistics & Probability Letters, Elsevier, vol. 45(1), pages 11-22, October.
    8. Cuevas, Antonio & Febrero, Manuel & Fraiman, Ricardo, 2006. "On the use of the bootstrap for estimating functions with functional data," Computational Statistics & Data Analysis, Elsevier, vol. 51(2), pages 1063-1074, November.
    9. Cattiaux, Patrick & Gozlan, Nathael, 2007. "Deviations bounds and conditional principles for thin sets," Stochastic Processes and their Applications, Elsevier, vol. 117(2), pages 221-250, February.
    10. Jan Mielniczuk & Małgorzata Wojtyś, 2010. "Estimation of Fisher information using model selection," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 72(2), pages 163-187, September.
    11. Mason David M. & Eubank Randy, 2012. "Moderate deviations and intermediate efficiency for lack-of-fit tests," Statistics & Risk Modeling, De Gruyter, vol. 29(2), pages 175-187, June.
    12. Ferraty, F. & Van Keilegom, I. & Vieu, P., 2012. "Regression when both response and predictor are functions," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 10-28.
    13. Dahmani, Abdelnasser & Ait Saidi, Ahmed & Bouhmila, Fatah & Aissani, Mouloud, 2009. "Consistency of the Tikhonov's regularization in an ill-posed problem with random data," Statistics & Probability Letters, Elsevier, vol. 79(6), pages 722-727, March.
    14. Menneteau, Ludovic, 2005. "Some laws of the iterated logarithm in Hilbertian autoregressive models," Journal of Multivariate Analysis, Elsevier, vol. 92(2), pages 405-425, February.


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