The non i.i.d. strong law of large numbers in 2-uniformly smooth Banach spaces

1984 ◽  
pp. 90-118 ◽  
Author(s):  
Bernard Heinkel
Keyword(s):  
Banach Spaces ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Uniformly Smooth ◽  
Large Numbers ◽  
Uniformly Smooth Banach Spaces

scholarly journals On Chung's strong law of large numbers in general Banach spaces

1988 ◽  
Vol 37 (1) ◽  
pp. 93-100 ◽  
Author(s):  
Bong Dae Choi ◽  
Soo Hak Sung
Keyword(s):  
Banach Spaces ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Real Numbers ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers

Let { Xn, n ≥ 1 } be a sequence of independent Banach valued random variables and { an, n, ≥ 1 } a sequence of real numbers such that 0 < an ↑ ∞. It is shown that, under the assumption with some restrictions on φ, Sn/an → 0 a.s. if and only if Sn/an → 0 in probability if and only if Sn/an → 0 in L1. From this result several known strong laws of large numbers in Banach spaces are easily derived.

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