Geodesic
Compact Law of the Iterated Logarithm for Matrix-Normalized Sums of Random Vectors
Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 4, pp. 752-767 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $(X_n)_{n\ge 1}$ be a sequence of independent centered random vectors in $R^d$. We give conditions under which the sequence $S_n=\sum_{i=1}^nX_i$ normalized by a matricial sequence $(H_n)$ satisfies a compact law of the iterated logarithm. As an application of this result, we obtain the compact law of the iterated logarithm for $B_n^{-1/2}S_n$ and for $\Delta_n^{-1/2}S_n$, where $B_n$ is the covariance matrix of $S_n$, and where $\Delta_n$ is the diagonal matrix whose $j$th diagonal term is the $j$th diagonal term of $B_n$; the eigenvalues of $B_n$ may go to infinity with different rates, but their iterated logarithms have to be equivalent.
Keywords: compact law of the iterated logarithm, matrix normings, sums of independent vectors. 
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A. Mokkadem; M. Pelletier. Compact Law of the Iterated Logarithm for Matrix-Normalized Sums of Random Vectors. Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 4, pp. 752-767. http://geodesic.mathdoc.fr/item/TVP_2007_52_4_a6/

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