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A central limit theorem for generalized multilinear forms

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  • de Jong, Peter

Abstract

Let X1, ..., Xn be independent random variables and define for each finite subset I [subset of] {1, ..., n} the [sigma]-algebra = [sigma]{Xi : i [epsilon] I}. In this paper -measurable random variables WI are considered, subject to the centering condition E(WI [short parallel] ) = 0 a.s. unless I [subset of] J. A central limit theorem is proven for d-homogeneous sums W(n) = [Sigma][short parallel]I[short parallel] = dWI, with var W(n) = 1, where the summation extends over all (nd) subsets I [subset of] {1, ..., n} of size [short parallel]I[short parallel] = d, under the condition that the normed fourth moment of W(n) tends to 3. Under some extra conditions the condition is also necessary.

Suggested Citation

  • de Jong, Peter, 1990. "A central limit theorem for generalized multilinear forms," Journal of Multivariate Analysis, Elsevier, vol. 34(2), pages 275-289, August.
    • Handle: RePEc:eee:jmvana:v:34:y:1990:i:2:p:275-289
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    Cited by:

    1. Gao, Jiti & Hong, Yongmiao, 2007. "Central limit theorems for weighted quadratic forms of dependent processes with applications in specification testing," MPRA Paper 11977, University Library of Munich, Germany, revised Dec 2007.
    2. Fan, Yanqin & Ullah, Aman, 1999. "Asymptotic Normality of a Combined Regression Estimator," Journal of Multivariate Analysis, Elsevier, vol. 71(2), pages 191-240, November.
    3. Kasprzak, Mikołaj J., 2020. "Stein’s method for multivariate Brownian approximations of sums under dependence," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4927-4967.
    4. Yuta Koike, 2023. "High-Dimensional Central Limit Theorems for Homogeneous Sums," Journal of Theoretical Probability, Springer, vol. 36(1), pages 1-45, March.
    5. Barton, N.H. & Etheridge, A.M. & Véber, A., 2017. "The infinitesimal model: Definition, derivation, and implications," Theoretical Population Biology, Elsevier, vol. 118(C), pages 50-73.
    6. Konrad Menzel, 2021. "Bootstrap With Cluster‐Dependence in Two or More Dimensions," Econometrica, Econometric Society, vol. 89(5), pages 2143-2188, September.
    7. Konrad Menzel, 2021. "Central Limit Theory for Models of Strategic Network Formation," Papers 2111.01678, arXiv.org.
    8. Robins, James M. & Li, Lingling & Tchetgen, Eric Tchetgen & van der Vaart, Aad, 2016. "Asymptotic normality of quadratic estimators," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3733-3759.
    9. Ivan Nourdin & Giovanni Peccati & Guillaume Poly & Rosaria Simone, 2016. "Classical and Free Fourth Moment Theorems: Universality and Thresholds," Journal of Theoretical Probability, Springer, vol. 29(2), pages 653-680, June.
    10. Ivan Nourdin & Giovanni Peccati & Xiaochuan Yang, 2022. "Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance," Journal of Theoretical Probability, Springer, vol. 35(3), pages 2020-2037, September.

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