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- Title
Strong Laws of Large Numbers for Double Sums of Banach Space Valued Random Elements.
- Authors
Parker, Robert; Rosalsky, Andrew
- 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 m a x 1 ≤ k ≤ m , 1 ≤ l ≤ n ∥ Σ i = 1 k Σ j = 1 l V i , j ∥ / b m , n → 0 a.c. and in ℒp as m ∨ n → ∞ is shown to hold if Σ m = 1 ∞ Σ n = 1 ∞ E ∥ V m , n ∥ p / b m , n p < ∞ . 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.
- Subjects
LAW of large numbers; BANACH spaces; ARITHMETIC mean
- Publication
Acta Mathematica Sinica, 2019, Vol 35, Issue 5, p583
- ISSN
1439-8516
- Publication type
Article
- DOI
10.1007/s10114-019-8058-5