Systematic and Unified Stochastic Tool to Determine the Multidimensional Joint Statistics of Arbitrary Partial Products of Ordered Random Variables

Abstract

In this paper, we introduce a systematic and unified stochastic tool to determine the joint statistics of partial products of ordered random variables (RVs). With the proposed approach, we can systematically obtain the desired joint statistics of any partial products of ordered statistics in terms of the Mellin transform and the probability density function in a unified way. Our approach can be applied when all the K-ordered RVs are involved, even for more complicated cases, for example, when only the Ks(Ks <; K) best RVs are also considered. As an example of their application, these results can be applied to the performance analysis of various wireless communication systems including wireless optical communication systems. For an applied example, we present the closed-form expressions for the exponential RV special case. We would like to emphasize that with the derived results based on our proposed stochastic tool, computational complexity and execution time can be reduced compared to the computational complexity and execution time based on an original multiple-fold integral expression of the conventional Mellin transform based approach which has been applied in cases of the product of RVs.

Keywords

• Joint PDF
• partial products
• Mellin transform (MT)
• order statistics
• probability density function (PDF)
• exponential random variables