Skip to main content
SpringerLink
Log in

Inequalities

  • Chapter
Asymptotics for Associated Random Variables

  • 981 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Azevedo, C., Oliveira, P.E.: Kernel-type estimation of bivariate distribution function for associated random variables. In: New Trends in Probability and Statistics, vol. 5. VSP, Utrecht (2000)

    Google Scholar 

  2. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)

    MATH  Google Scholar 

  3. Birkel, T.: Moment bounds for associated sequences. Ann. Probab. 16, 1184–1193 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Birkel, T.: On the convergence rate in the central limit theorem for associated processes. Ann. Probab. 16, 1685–1698 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bulinski, A.: On the convergence rates in the CLT for positively and negatively dependent random fields. In: Ibragimov, I., Zaitov, Y. (eds.) Probability Theory and Mathematical Statistics, pp. 3–14. Gordon & Breach, Amsterdam (1996)

    Google Scholar 

  6. Cai, Z., Roussas, G.G.: Efficient estimation of a distribution function under quadrant dependence. Scand. J. Stat. 25, 211–224 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cox, J., Grimmett, G.: Central limit theorems for associated random variables and the percolation model. Ann. Probab. 12, 514–528 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dewan, I., Prakasa Rao, B.L.S.: A general method of density estimation for associated random variables. J. Nonparametr. Stat. 10, 405–420 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Henriques, C., Oliveira, P.E.: Estimation of a two-dimensional distribution function under association. J. Stat. Plan. Inference 113, 137–150 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Henriques, C., Oliveira, P.E.: Exponential rates for kernel density estimation under association. Stat. Neerl. 59, 448–466 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Henriques, C., Oliveira, P.E.: Convergence rates for the strong law of large numbers under association. Technical report, Pré-Publicações do Departamento de Matemática da Universidade de Coimbra 08-14 (2008)

    Google Scholar 

  12. Ioannides, D., Roussas, G.G.: Exponential inequality for associated random variables. Stat. Probab. Lett. 42, 423–431 (1998)

    Article  MathSciNet  Google Scholar 

  13. Lebowitz, J.: Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems. Commun. Math. Phys. 28, 313–321 (1972)

    Article  MathSciNet  Google Scholar 

  14. Louhichi, S.: Rosenthal’s inequality for LPQD sequences. Stat. Probab. Lett. 42, 139–144 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Masry, E.: Multivariate probability density estimation for associated processes: strong consistency and rates. Stat. Probab. Lett. 58, 205–219 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Newman, C.: Normal fluctuations and the FKG inequalities. Commun. Math. Phys. 74, 119–128 (1980)

    Article  MATH  Google Scholar 

  17. Newman, C.: A general central limit theorem for FKG systems. Commun. Math. Phys. 91, 75–80 (1983)

    Article  MATH  Google Scholar 

  18. Newman, C.: Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong, Y. (ed.) Inequalities in Statistics and Probability, vol. 5, pp. 127–140. Inst. Math. Stat., Hayward (1984)

    Chapter  Google Scholar 

  19. Newman, C., Wright, A.: An invariance principle for certain dependent sequences. Ann. Probab. 9, 671–675 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  20. Newman, C., Wright, A.: Associated random variables and martingale inequalities. Z. Wahrscheinlichkeitstheor. Verw. Geb. 59, 361–371 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  21. Oliveira, P.E.: An exponential inequality for associated variables. Stat. Probab. Lett. 73, 189–197 (2005)

    Article  MATH  Google Scholar 

  22. Prakasa Rao, B.L.S.: Bernstein-type inequality for associated sequences. In: Ghosh, J., Mitra, S., Parthasarathy, K., Prakasa Rao, B.L.S. (eds.) Statistics and Probability: A Raghu Raj Bahadur Festschrift, pp. 499–509. Wiley Eastern, New Delhi (1993)

    Google Scholar 

  23. Shao, Q.M., Yu, H.: Weak convergence for weighted empirical processes of dependent sequences. Ann. Probab. 24, 2098–2127 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  24. Sung, S.: A note on the exponential inequality for associated random variables. Stat. Probab. Lett. 77, 1730–1736 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Xing, G., Yang, S.: Notes on the rate of strong convergence for associated random variables. Commun. Stat., Theory Methods 38, 1–4 (2009)

    MathSciNet  Google Scholar 

  26. Xing, G., Yang, S., Liu, A.: Exponential inequalities for positively associated random variables and applications. J. Inequal. Appl. 2008, 385362 (2008)

    Article  MathSciNet  Google Scholar 

  27. Yang, S.: Moment inequalities for partial sums of random variables. Sci. China Ser. A 44, 1–6 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  28. Yang, S., Su, C., Yu, K.: A general method to the strong law of large numbers and its applications. Stat. Probab. Lett. 78, 794–803 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Yu, H.: A Glivenko–Cantelli lemma and weak convergence for empirical processes of associated sequences. Probab. Theory Relat. Fields 95, 357–370 (1993)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, CMUC, University of Coimbra, Coimbra, Portugal

    Paulo Eduardo Oliveira

Authors
  1. Paulo Eduardo Oliveira
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Oliveira, P.E. (2012). Inequalities. In: Asymptotics for Associated Random Variables. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25532-8_2

Download citation

Publish with us

Policies and ethics

Search

Navigation