Sub-Independence: An Expository Perspective

Abstract

Limit theorems as well as other well-known results in probability and statistics are often based on the distribution of the sums of independent random variables. The concept of sub-independence, which is much weaker than that of independence, is shown to be sufficient to yield the conclusions of these theorems and results. It also provides a measure of dissociation between two random variables which is much stronger than uncorrelatedness.

Keywords:

• Characteristics function
• Independence
• Limit theorems
• Sub-independence

Mathematics Subject Classification:

• Primary 60E10
• Secondary 60E05

Acknowledgments

The author is grateful to Barry Arnold for his numerous communications regarding this work, in particular his elegant proof of (1.7) and his suggestion to construct Example 2.8. Thanks go to Hans Volkmer for his invaluable suggestion concerning the definition of sub-independent continuous random variables in terms of events. Thanks also go to Adel Mohammadpour for his careful reading of the first draft of the article. Finally, we are also grateful to the referees for their constructive suggestions improving the presentation of the content of this article.