Abstract
For a sequence of independent identically distributed Euclidean random vectors, we prove a compact Law of the iterated logarithm when finitely many maximal terms are omitted from the partial sum. With probability one, the limiting cluster set of the appropriately operator normed partial sums is the closed unit Euclidean ball. The result is proved under the hypotheses that the random vectors belong to the Generalized Domain of Attraction of the multivariate Gaussian law and satisfy a mild integrability condition. The integrability condition characterizes how many maximal terms must be omitted from the partial sum sequence.
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REFERENCES
Araujo, and Gine, E. (1980). The Central Limit Theorem for Real andBanach Valued Random Variables, Wiley, New York.
Billingsley, P. (1966). Convergence of types in k-space. Z. Wahrsh. Verw. Gebiete. 5, 175–179.
Chung, K. L. (1974). A Course in Probability Theory, Academic Press, New York.
deAcosta, A. (1981). Inequalities for B-Valued random vectors with applications to the strong law of large numbers. Ann Probab. 9, 157–161.
Feller, W. (1968). An extension of the law of the iterated logarithm to variables without variance. J. Math. Mechan. 18, 343–355.
Hahn, M. G., and Klass, M. J. (1980). Matrix normalization of sums of random vectors in the domain of attraction of the multivariate normal. Ann. Probab. 8, 262–280.
Hahn, M. G., and Klass, M. J. (1980). The generalized domain of attraction of spherically symmetric stable laws on ℝd. Proceedings of the Conference on Probability in Vector Spaces II, Poland, Lecture Notes in Math. 828, 52–81.
Kuelbs, J., and Ledoux M. (1987). Extreme values and the law of the iterated logarithm. Prob. Theory and Rel. Fields 74, 319–340.
Ledoux, M., and Talagrand, M. (1991). Probability in Banach Spaces, Springer Verlag, Berlin.
Meerschaert, M. (1988). Regular variation in ℝk. Proc. Amer. Math. Soc. 102, 341–348.
Pruitt, W. E. (1981). General one sided laws of the iterated logarithm. Ann. Probab. 9, 1–48.
Sepanski, S. (1999). A compact law of the iterated logarithm for random vectors in the generalized domain of attraction of the multivariate gaussian law. J. Theoret. Probab. 12, 757–778.
Weiner, D. C. (1986). A law of the iterated logarithm for distributions in the generalized domain of attraction of a nondegenerate gaussian law. Prob. Theory andRel. Fields 72, 337–357.
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Sepanski, S.J. Extreme Values and the Multivariate Compact Law of the Iterated Logarithm. Journal of Theoretical Probability 14, 989–1018 (2001). https://doi.org/10.1023/A:1017576903851
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DOI: https://doi.org/10.1023/A:1017576903851