Some results following from conditional characteristic functions

ABSTRACT

The purpose of this article is to derive rigorously some results following from conditional characteristic functions and anticipate that they will prove to be of significant applicability. Specifically, Khintchine weak law of large numbers, Lévy central limit theorem, inversion theorem, uniqueness theorem, characterization of identical distribution, and characterization of independence are all generalized to conditional setup.

Keywords:

• Conditional central limit theorem
• Conditional characteristic function
• Conditionally identical distribution
• Conditional independence
• Conditional inversion theorem
• Conditional uniqueness theorem
• Conditional weak law of large numbers.

Mathematics Subject Classification:

• Primary 60E10
• 60F05
• Secondary 60G50

Acknowledgments

The authors would like to express their deep gratitude to the anonymous referees for their valuable comments and suggestions, which resulted in the present version of this paper.

Funding

This work was supported by National Natural Science Foundation of China (No. 11101452), Natural Science Foundation Project of CQ CSTC of China (No. 2012jjA00035), and Natural Science Foundation Project of CTBU of China (No. 1352001).