On complete convergence in mean for double sums of independent random elements in Banach spaces
Tóm tắt
For a double array of random elements {T
m,n
, m ≥ 1, n ≥ 1} in a real separable Banach space X, we study the notion of T
m,n
converging completely to 0 in mean of order p where p is a positive constant. This notion is stronger than (i) T
m,n
converging completely to 0 and (ii) T
m,n
converging to 0 in mean of order p as max{m, n} →∞. When X is of Rademacher type p (1 ≤ p ≤ 2), for a double array of independent mean 0 random elements {V
m,n
, m ≥ 1, n ≥ 1} in X and a double array of constants {b
m,n
, m ≥ 1, n ≥ 1}, conditions are provided under which max1≤k≤m,1≤l≤n||Ʃi=1
kƩj=1
l
V
i,j||/b
m,n
converges completely to 0 in mean of order p. Moreover, these conditions are shown to provide an exact characterization of Rademacher type p (1 ≤ p ≤ 2) Banach spaces. Illustrative examples are provided.
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