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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 Lp for conditionally independent and conditionally identically distributed random variables are established, respectively.

EXTENSIONS OF SEVERAL CLASSICAL RESULTS FOR INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM VARIABLES TO CONDITIONAL CASES