DOI QR코드

DOI QR Code

A WEAK LAW FOR WEIGHTED SUMS OF ARRAY OF ROW NA RANDOM VARIABLES

  • Baek, Jong-Il (School of Mathematics & Informational Statistics, and Institute of Basic Natural Sciences, Wonkwang University) ;
  • Liang, Han-Ying (Department of Applied Mathematics, Tongji University) ;
  • Choi, Jeong-Yeol (School of Mathematics & Informational Statistics, and Institute of Basic Natural Sciences, Wonkwang University)
  • Published : 2003.05.01

Abstract

Let {$x_{nk}\;$\mid$1\;\leq\;k\;\leq\;n,\;n\;\geq\;1$} be an array of random varianbles and $\{a_n$\mid$n\;\geq\;1\}\;and\;\{b_n$\mid$n\;\geq\;1} be a sequence of constants with $a_n\;>\;0,\;b_n\;>\;0,\;n\;\geq\;1. In this paper, for array of row negatively associated(NA) random variables, we establish a general weak law of large numbers (WLLA) of the form (${\sum_{\kappa=1}}^n\;a_{\kappa}X_{n\kappa}\;-\;\nu_{n\kappa})\;/b_n$ converges in probability to zero, as $n\;\rightarrow\;\infty$, where {$\nu_{n\kappa}$\mid$1\;\leq\;\kappa\;\leq\;n,\;n\;\geq\;1$} is a suitable array of constants.

Keywords

negatively associated random variables;weak law of large numbers;weighted sum

References

  1. Statist. Probab. Lett. v.15 A note on the almost sure convergence of sums of negatively dependent random variables P.Matula https://doi.org/10.1016/0167-7152(92)90191-7
  2. J. Math. and Math. Sci. v.14 On the weak law of large number for normed weighted sums of i.i.d random variables, Internet A.Adler;A.Rosalsky https://doi.org/10.1155/S0161171291000182
  3. Ann. Statist. v.11 Negative association of random variables with applications K.Joag-Dev;F.Proschan https://doi.org/10.1214/aos/1176346079
  4. Amer. J. Math. v.68 A limit theorem for random variables with in finite moments W.Feller https://doi.org/10.2307/2371837
  5. Theory and application of infinite series(2nd English ed.) K.Knopp
  6. Ann. Math. Statist v.37 Some concepts of dependence E.L.Lehmann https://doi.org/10.1214/aoms/1177699260
  7. Probability Theory: Independence, Interchangeability, Martingales(2nd ed.) Y.S.Chow;H.Teicher
  8. Statist. Probab. Lett. v.22 On the weak law of large number for arrays D.H.Hong;K.S.Oh https://doi.org/10.1016/0167-7152(94)00047-C
  9. Stochastic Anal. Appl. v.5 Some general strong laws for weighted sums of stochastically dominated random variables A.Adler;A.Rosalsky https://doi.org/10.1080/07362998708809104
  10. Statist. Probab. Lett. v.14 The weak law of large numbers for arrays A.Gut https://doi.org/10.1016/0167-7152(92)90209-N
  11. Probability Theory I(4th ed.) M.Loeve
  12. Statist. Probab. Lett. v.38 Weak law of large numbers for arrays S.H.Sung https://doi.org/10.1016/S0167-7152(97)00159-4
  13. J. Multivariate Anal. v.37 A weak law for normed weighted sums of random elements in Rademacher Type p Banach spaces A.Adler;A.A.Rosalsky;R.L.Taylor https://doi.org/10.1016/0047-259X(91)90083-E
  14. Ann. Univ. Mariae Curie-sklodowska Sect. v.A 51 On the weak law of large numbers for randomly indexed partial sums for arrays P.Kowalski;Z.Rychlik
  15. Comm. Statist v.A 10 Positive dependence in multivariate distributions K.Alam;K.M.Lal Saxena