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Central Limit Theorems revisited

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  • Kundu, Subrata
  • Majumdar, Suman
  • Mukherjee, Kanchan

Abstract

A Central Limit Theorem for a triangular array of row-wise independent Hilbert-valued random elements with finite second moment is proved under mild convergence requirements on the covariances of the row sums and the Lindeberg condition along the evaluations at an orthonormal basis. A Central Limit Theorem for real-valued martingale difference arrays is obtained under the conditional Lindeberg condition when the row sums of conditional variances converge to a (possibly degenerate) constant. This result is then extended, first to multi-dimensions and next to Hilbert-valued elements, under appropriate convergence requirements on the conditional and unconditional covariances and the conditional Lindeberg condition along (orthonormal) basis evaluations. Extension to include Banach- (with a Schauder basis) valued random elements is indicated.

Suggested Citation

  • Kundu, Subrata & Majumdar, Suman & Mukherjee, Kanchan, 2000. "Central Limit Theorems revisited," Statistics & Probability Letters, Elsevier, vol. 47(3), pages 265-275, April.
    • Handle: RePEc:eee:stapro:v:47:y:2000:i:3:p:265-275
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    References listed on IDEAS

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    1. Basu, A. K., 1988. "Uniform and nonuniform estimates in the CLT for Banach valued dependent random variables," Journal of Multivariate Analysis, Elsevier, vol. 25(2), pages 153-163, May.
    2. Xie, Yingchao, 1995. "Limit theorems of Hilbert valued semimartingales and Hilbert valued martingale measures," Stochastic Processes and their Applications, Elsevier, vol. 59(2), pages 277-293, October.
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    2. Jiménez-Gamero, M.D. & Alba-Fernández, M.V. & Jodrá, P. & Barranco-Chamorro, I., 2017. "Fast tests for the two-sample problem based on the empirical characteristic function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 137(C), pages 390-410.
    3. Baringhaus, Ludwig & Gaigall, Daniel, 2023. "A goodness-of-fit test for the compound Poisson exponential model," Journal of Multivariate Analysis, Elsevier, vol. 195(C).
    4. Apostolos Batsidis & María Dolores Jiménez-Gamero & Artur J. Lemonte, 2020. "On goodness-of-fit tests for the Bell distribution," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(3), pages 297-319, April.
    5. Joydeep Chowdhury & Probal Chaudhuri, 2020. "Convergence rates for kernel regression in infinite-dimensional spaces," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(2), pages 471-509, April.
    6. Bruno Ebner & Norbert Henze & Simos Meintanis, 2024. "A unified approach to goodness-of-fit testing for spherical and hyperspherical data," Statistical Papers, Springer, vol. 65(6), pages 3447-3475, August.
    7. Norbert Henze & Jaco Visagie, 2020. "Testing for normality in any dimension based on a partial differential equation involving the moment generating function," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(5), pages 1109-1136, October.
    8. 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.
    9. Gupta, A. K. & Henze, N. & Klar, B., 2004. "Testing for affine equivalence of elliptically symmetric distributions," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 222-242, February.
    10. Norbert Henze & María Dolores Jiménez‐Gamero, 2021. "A test for Gaussianity in Hilbert spaces via the empirical characteristic functional," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(2), pages 406-428, June.
    11. Delgado, Miguel A. & Song, Xiaojun, 2018. "Nonparametric tests for conditional symmetry," Journal of Econometrics, Elsevier, vol. 206(2), pages 447-471.
    12. Krebs, Johannes T.N., 2019. "The bootstrap in kernel regression for stationary ergodic data when both response and predictor are functions," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 620-639.
    13. G. I. Rivas-Martínez & M. D. Jiménez-Gamero & J. L. Moreno-Rebollo, 2019. "A two-sample test for the error distribution in nonparametric regression based on the characteristic function," Statistical Papers, Springer, vol. 60(4), pages 1369-1395, August.
    14. M. Jiménez Gamero, 2014. "On the empirical characteristic function process of the residuals in GARCH models and applications," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 409-432, June.
    15. Norbert Henze & Pierre Lafaye De Micheaux & Simos G. Meintanis, 2022. "Tests for circular symmetry of complex-valued random vectors," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(2), pages 488-518, June.
    16. Shishebor, Z. & Soltani, A.R. & Zamani, A., 2011. "Asymptotic distribution for periodograms of infinite dimensional discrete time periodically correlated processes," Journal of Multivariate Analysis, Elsevier, vol. 102(7), pages 1118-1125, August.
    17. Rajeshwari Majumdar & Suman Majumdar, 2023. "Equivalence of Asymptotic Normality of the Two Sample Pivot and the Vector of Standardized Sample Means," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1684-1707, August.
    18. Jiménez-Gamero, M. Dolores & Kim, Hyoung-Moon, 2015. "Fast goodness-of-fit tests based on the characteristic function," Computational Statistics & Data Analysis, Elsevier, vol. 89(C), pages 172-191.
    19. Norbert Henze & Stefan Koch, 2020. "On a test of normality based on the empirical moment generating function," Statistical Papers, Springer, vol. 61(1), pages 17-29, February.
    20. Jiménez-Gamero, M.D. & Alba-Fernández, M.V., 2019. "Testing for the Poisson–Tweedie distribution," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 164(C), pages 146-162.
    21. Petr Čoupek & Viktor Dolník & Zdeněk Hlávka & Daniel Hlubinka, 2024. "Fourier approach to goodness-of-fit tests for Gaussian random processes," Statistical Papers, Springer, vol. 65(5), pages 2937-2972, July.
    22. Clémençon, Stephan & Huet, Nathan & Sabourin, Anne, 2024. "Regular variation in Hilbert spaces and principal component analysis for functional extremes," Stochastic Processes and their Applications, Elsevier, vol. 174(C).

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