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

Published online by Cambridge University Press:  17 April 2009

Bong Dae Choi  and
Soo Hak Sung
Bong Dae Choi
Affiliation:
Department of Applied Mathematics, Korea Advanced Institute of Science and TechnologyP.O.Box 150, Cheongryang, Seoul, Korea
Soo Hak Sung
Affiliation:
Department of Applied Mathematics, Korea Advanced Institute of Science and TechnologyP.O.Box 150, Cheongryang, Seoul, Korea
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Abstract

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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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 37 , Issue 1 , February 1988 , pp. 93 - 100
Copyright
Copyright © Australian Mathematical Society 1988

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