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Inequalities

In: Asymptotics for Associated Random Variables

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  • Paulo Eduardo Oliveira

    (University of Coimbra, Department of Mathematics, CMUC)

Abstract

This chapter sets the basic tools to prove the asymptotic results that are to come in the following chapters. The first three sections are concerned with different types of inequalities on joint distributions of associated random variables and moments of sums. It is interesting that, although association is defined with a somewhat vague requirement, it is possible to recover versions for moment inequalities which are quite close to the independent case, thus paving the way to find asymptotic results that are also similar to the ones found in the independence framework. One of the key issues with association is the ability to control joint distributions from the marginal distributions using the covariance structure of the random variables. This is explored mainly in Sects. 2.5 and 2.6. The inequalities proved in these sections will provide the means to use the coupling technique, common to prove convergence results. Sect. 2.5 shows that at least the convergence in distribution is concerned with the covariance structure that completely describes the behaviour of associated variables. This chapter is a fundamental one for the remaining text.

Suggested Citation

  • Paulo Eduardo Oliveira, 2012. "Inequalities," Springer Books, in: Asymptotics for Associated Random Variables, chapter 0, pages 35-66, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-25532-8_2
    DOI: 10.1007/978-3-642-25532-8_2
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