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Franck, W. E. and Hanson, D. L. , Some results giving rates of convergence in the law of large numbers for weighted sums of independent variables,. Trans. Amer. Math. Soc. , 124 (1966), 347–359. MR 33 #8017

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Abstract  

We study necessary and sufficient conditions for the almost sure convergence of averages of independent random variables with multidimensional indices obtained by certain summability methods.

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Let X 1, X 2,… be independent, but not necessarily identically distributed random variables in the domain of attraction of a stable law with index 0<a<2. This paper uses M n=max 1 ? i ? n|X i| to establish a self-normalized law of the iterated logarithm (LIL) for partial sums. Similarly self-normalized increments of partial sums are studied as well. In particular, the results of self-normalized sums of Horváth and Shao[9]under independent and identically distributed random variables are extended and complemented. As applications, some corresponding results for self-normalized weighted sums of iid random variables are also concluded.

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] Cuesta , J. A. Matrán , C. 1988 Strong convergence of weighted sums of random elements through the equivalence of sequences of distributions J. Multivariate Analysis

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iterated logarithm for weighted sums of random variables, Ann. Probab . 12 (1984), 35-44. MR 85d :60064 A note on the law of the iterated logarithm for weighted sums of random variables Ann. Probab

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overall prestige as the normalized weighted sum of the contribution of each one journal to the overall prestige as follows: Definition 1 A measure of the overall prestige R of journals with ranking score above a threshold z , for a

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normalized weighted sum of the article contribution to the overall prestige as follows: Definition 1 Given a configuration of dimension-specific scores of size n × d , and a 1 × d vector of dimension

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the multidimensional prestige of influential fields at the university as the normalized weighted sum of the field contribution to the overall prestige as follows: Definition 1 Given a configuration of dimension

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or SS PIR ): the weighted sum of publications authored by the scientist, the weights for each publication being equal to the quality index of the publication (PII or PIR). Fractional Scientific

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