Strong Laws of Large Numbers for Double Sums of Banach Space Valued Random Elements |
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Authors: | Parker Robert Rosalsky Andrew |
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Affiliation: | 1.Department of Biostatistics, University of Florida, Gainesville, Florida 32611-7450, USA;2.Department of Statistics, University of Florida, Gainesville, Florida 32611-8545, USA |
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Abstract: | For a double array {Vm,n, m≥1,n≥1} of independent, mean 0 random elements in a real separable Rademacher type p (1≤p≤ 2) Banach space and an increasing double array {bm,n, m≥1,n ≥ 1} of positive constants, the limit law max1≤k≤m,1≤l≤n||Σ i=1k||Σ j=1l Vi,j||/bm,n → 0 a.c. and in Lp as m ∨ n → ∞ is shown to hold if Σm=1∞ Σn=1∞ E||Vm,n||p/bm,np < ∞. This strong law of large numbers provides a complete characterization of Rademacher type p Banach spaces. Results of this form are also established when 0<p≤1 where no independence or mean 0 conditions are placed on the random elements and without any geometric conditions placed on the underlying Banach space. |
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Keywords: | Real separable Banach space double array of independent random elements strong law of large numbers almost sure convergence Rademacher type p Banach space convergence in Lp |
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