From Conditional Independence to Conditionally Negative Association: Some Preliminary Results

Abstract

Conditional moment estimates on the cumulative sum of conditionally independent random variables are derived, conditional prophet inequalities for conditionally independent random variables are established, a comparison theorem on conditional moment inequalities between conditionally independent and conditionally negatively associated random variables is established. As applications of these results, a conditional Rosenthal type inequality and two conditional Kolmogorov exponential inequalities for conditionally negatively associated random variables are obtained.

The Conditional Borel-Cantelli Lemma and Applications

“…For instance, Majerek et al [13] for conditional strong law of large number, Yuan et al [22,23] for a conditional version of the classical central limit theorem, a conditional version of the extended Kolmogorov-Feller weak law of large numbers, Christofides and Hadjikyriakou [7] for conditional demimartingale, Liu and Prakasa Rao [11] for conditional Borel-Cantelli lemma, Ordonez Cabrera et al [15] for conditionally negatively quadrant dependence, Wang and Wang [21] for conditional demimartingale and conditional N-demimartingale, Yuan and Lei (2013) for conditionally strong mixing.…”