Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A CENTRAL LIMIT THEOREM FOR GENERAL WEIGHTED SUMS OF LPQD RANDOM VARIABLES AND ITS APPLICATION

  • Ko, Mi-Hwa (Statistical Research Center for Complex Systems Seoul National University) ;
  • Kim, Hyun-Chull (Department of Mathematics Education Daebul University) ;
  • Kim, Tae-Sung (Department of Mathematics and Institute of Basic Science WonKwang University)
  • Published : 2006.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we derive the central limit theorem for ${\sum}^n_{i=l}\;a_{ni}{\xi}_{i},\;where\;\{a_{ni},\;1\;{\le}\;i\;{\le}n\}$ is a triangular array of non-negative numbers such that $sup_n{\sum}^n_{i=l}\;a^2_{ni}\;<\;{\infty},\;max_{1{\le}i{\le}n\;a_{ni}{\to}\;0\;as\;n{\to}{\infty}\;and\;{\xi}'_{i}s$ are a linearly positive quadrant dependent sequence. We also apply this result to consider a central limit theorem for a partial sum of a generalized linear process of the form $X_n\;=\;{\sum}^{\infty}_{j=-{\infty}}a_{k+j}{\xi}_{j}$.

Keywords

References

  1. P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968
  2. T. Birkel, A functional central limit theorem for positively dependent random vari- ables, J. Multivariate Anal. 44 (1993) 314-320 https://doi.org/10.1006/jmva.1993.1018
  3. J. T. Cox and G. Grimmett, Central limit theorems for associated random variables and the percolation model, Ann. Probab. 12 (1984), no. 2, 514-528 https://doi.org/10.1214/aop/1176993303
  4. J. Esary, F. Proschan, and D. Walkup, Association of random variables with ap- plications, Ann. Math. Statist. 38 (1967), 1466-1474 https://doi.org/10.1214/aoms/1177698701
  5. I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Volters, Groningen, 1971
  6. T. S. Kim and J. L. Baek, A central limit theorem for stationary linear processes generated by linearly positive quadrant dependent process, Statist. Probab. Lett. 51 (2001), no. 3, 299-305 https://doi.org/10.1016/S0167-7152(00)00168-1
  7. E. L. Lehmann, Some concepts of dependence, Ann. Math. Statist. 37 (1966), 1137-1153 https://doi.org/10.1214/aoms/1177699260
  8. C. M. Newman, Normal fluctuations and the FKG inequalities, Comm. Math. Phys. 91 (1980), 75-80
  9. C. M. Newman, Asymptotic independence and limit theorems for positively and negatively dependent random variables, In, Inequalities in Statistics and Probability (Y. L. Tong, Ed.), IMS Lecture Notes-Monogragh Series, 5 (1984), 127-140 (Institute of Mathematical Statistics, Hayward, California)
  10. V. V. Petrov, Sums of Independent Random Variables, Berlin, Heidelberg, New York, 1975

Cited by

  1. A central limit theorem for weighted sums of associated random field vol.45, pp.1, 2016, https://doi.org/10.1080/03610926.2013.815212