The essential dependence for a group of random vectors

Abstract

Considering random variables $X_i,i=1,2,\dots ,n$ which are from dominated distributions, we divide them into $\{{\mathit{X}}^{(1)},\dots ,{\mathit{X}}^{(m)}\},m\le n,$ where ${\mathit{X}}^{(j)},j=1,\dots ,m$ are random vectors. Inspired by copula and Kullback-Leibler divergence, by extending probability density function to Radon-Nikodym derivative w.r.t. a σ-finite product measure, the amount $\mathit{DivM}({\mathit{X}}^{(1)},\dots ,{\mathit{X}}^{(m)})$ is proposed with some desirable properties to describe the essential dependence for that group of random vectors. Some examples are given to demonstrate the amount can be applied to describe the essential dependence under both continuous and discrete distributions and can capture the associations such as MTP₂, POD.

Keywords:

• Copula
• the essential dependence
• random vector
• Radon-Nikodym derivative
• Kullback-Leibler divergence

2010 MSC:

• 62H20
• 62B86
• 60A10
• 94A17

Acknowledgments

This work is dedicated to the memory of Professor Lixin Song who proposed this amount of essential dependence combined with Copula and Kullback-Leibler divergence and provided the key point in the generalization of probability density function.

Additional information

Funding

The first author is sponsored by the China Scholarship Council. The second author is supported by the National Natural Sciences Foundation of China (Grant No. 11571058), the Fundamental Research Funds for the Central Universities (Grant No. DUT18LK18), the Dalian High Level Talent Innovation Programme (Grant No. 2015R051) and the High-level innovative and entrepreneurial talents support plan in Dalian (Grant No. 2017RQ041). The third author is supported by the National Natural Sciences Foundation of China (Grant No. 11471065).