• Journal of the Korean Mathematical Society
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• v.52 no.2
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• pp.431-445
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• 2015

Extensions of the Kolmogorov convergence criterion and the Marcinkiewicz-Zygmund inequalities from independent random variables to conditional independent ones are derived. As their applications, a conditional version of the Marcinkiewicz-Zygmund strong law of large numbers and a result on convergence in $L^p$ for conditionally independent and conditionally identically distributed random variables are established, respectively.

• Journal of the Korean Mathematical Society
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• v.53 no.6
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• pp.1275-1292
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• 2016

The Marcinkiewicz-Zygmund strong law of large numbers for conditionally independent and conditionally identically distributed random variables is an existing, but merely qualitative result. In this paper, for the more general cases where the conditional order of moment belongs to (0, ${\infty}$) instead of (0, 2), we derive results on convergence rates which are quantitative ones in the sense that they tell us how fast convergence is obtained. Furthermore, some conditional probability inequalities are of independent interest.