A Strong Law of Large Numbers in Banach Spaces of Type Φ

1987 ◽  
pp. 89-97 ◽  
Author(s):  
István Fazekas
Keyword(s):  
Banach Spaces ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Large Numbers

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.

scholarly journals On conditions for the strong law of large numbers in general Banach spaces

2000 ◽  
Vol 24 (1) ◽  
pp. 29-42
Author(s):  
Anna Kuczmaszewska ◽  
Dominik Szynal
Keyword(s):  
Banach Spaces ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Large Numbers

We give Chung-Teicher type conditions for the SLLN in general Banach spaces under the assumption that the weak law of large numbers holds. An example is provided showing that these conditions can hold when some earlier known conditions fail.

scholarly journals Best Possible Sufficient Conditions for Strong Law of Large Numbers for Multi-Indexed Orthogonal Random Elements

2007 ◽  
Vol 2007 ◽  
pp. 1-15
Author(s):  
Kuo-Liang Su
Keyword(s):  
Banach Spaces ◽  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
Random Elements

It will be shown and induced that thed-dimensional indices in the Banach spaces version conditions∑n(E‖Xn‖p/|nα|p)<∞are sufficient to yieldlimmin1≤j≤d(nj)→∞(1/|nα|)∑k≤n∏j=1d(1−(kj−1)/nj)Xk=0a.s. for arrays of James-type orthogonal random elements. Particularly, it will be shown also that there are the best possible sufficient conditions for multi-indexed independent real-valued random variables.

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