Conditional demimartingales and related results

Abstract

Conditional N-demimartingales Maximal inequalities Moment inequalities Strong law of large numbers Conditional complete convergence a b s t r a c tIn this paper we deal with the classes of F -demimartingales and conditional N-demimartingales and for these new classes of random objects we provide a number of maximal and moment inequalities as well as related asymptotic results.

From Conditional Independence to Conditionally Negative Association: Some Preliminary Results

“…according to Lemma 20 of Christofides and Hadjikyriakou (2013) and the definition of conditional median of ξ .…”

“…Starting from the conditional independence given a σ -algebra F, the past nearly a decade has witnessed the active development of a rich probability theory of conditional dependence and many important theoretical results have been obtained. See, for instance, Christofides and Hadjikyriakou (2013) for conditional demimartingale, Liu and Prakasa Rao (2013) for conditional Borel-Cantelli lemma, Ordóñez Cabrera et al (2012) for conditionally negatively quadrant dependence, Wang and Wang (2013) for conditional demimartingale and conditional N-demimartingale, Yuan and Lei (2013) for conditionally strong mixing, Yuan and Xie (2012) for conditionally linearly negatively quadrant dependence, and Yuan and Yang (2011) for conditional association. These motivate our original interest in conditionally independent and conditionally negatively associated random variables.…”

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.…”

Convergence Rates for Sequences of Conditionally Independent and Conditionally Identically Distributed Random Variables

“…See, for instance, Christofides and Hadjikyriakou [4] for conditional demimartingale, Liu and Prakasa Rao [9] for conditional Borel-Cantelli lemma, Ordóñez Cabrera et al [12] for conditionally negatively quadrant dependence, Wang and Wang [16] for conditional demimartingale and conditional N-demimartingale, Yuan and Xie [22] for conditionally linearly negatively quadrant dependence, Yuan and Lei [17] for conditional strong mixing. These achievements also stimulate us to study conditionally independent and conditionally identically distributed random variables.…”