Sains Malaysiana 45(7)(2016): 11771181

On P-Convergence of Four Dimensional Weighted Sums of Double Random Variables

(Hasil Tambah Berpemberat Empat Dimensi Berganda Pemboleh Ubah Rawak ke atas Penumpuan-P)

 

RICHARD F. PATTERSON1 & EKREM SAVAS2*

 

1Department of Mathematics and Statistics, University of North Florida Jacksonville

Florida, 32224, USA

 

2Department of Mathematics, Istanbul Ticaret University, Sutluce-Istanbul, Turkey

 

Diserahkan: 6 Oktober 2015/Diterima: 19 Disember 2015

 

 

ABSTRACT

The goal of this paper was to present a series of limit theorems that characterizes independent double random variables via four dimensional summability transformation. In order to accomplish this goal we began with the presentation of the following theorem that characterize pairwise independent random variables: let [xk,l] be a double sequence of pairwise independent random variables such that [xk, l] was uniformly integrable. Let [am, n, k, l] be a four dimensional matrix such that C for all ordered pair (m, n) and for some C and converges to 0 in probability. Then (xk,l E(xk,l) converges in mean to 0. Other extensions and variations via multidimensional transformation shall also be presented.

 

Keywords: Double sequences Pringsheim limit point; P-convergent; RH-Regular

 

ABSTRAK

Penyelidikan ini bertujuan untuk membentangkan satu siri teorem had yang mencirikan pemboleh ubah rawak bebas berganda melalui keterhasiltambahan transformasi empat dimensi. Untuk mencapai matlamat ini, kami mulakan dengan memberikan teorem yang mencirikan pasangan demi pasangan pemboleh ubah rawak: biar [xk,l] menjadi jujukan ganda dua pasangan demi pasangan pemboleh ubah rawak bebas supaya [xk, l] menjadi seragam terkamir. Biar [am, n, k, l] menjadi empat dimensi matriks supaya C untuk semua pasangan yang disusun (m, n) dan bagi sesetengah C dan penumpuan dalam kebarangkalian kepada 0. Kemudian (xk,l E(xk,l) menumpu pada min untuk 0. Perluasan lain dan variasi melalui transformasi bermultimatra turut dikemukakan.

 

Kata kunci: Jujukan ganda dua titik had Pringsheim; penumpuan P; RH biasa

RUJUKAN

Hamilton, H.J. 1936. Transformations of Multiple Sequences. Duke Math. Jour. 2: 29-60.

Patterson, R.F. 2000. Analogues of some fundamental theorems of summability theory, Internat. J. Math. & Math. Sci. 23(1): 1-9.

Patterson, R.F. & Savas, E. 2013. On double sequences of continuous functions having continuous P-limits. Publ. Math. Debrecen 82(1): 43-58.

Patterson, R.F. & Savas, E. 2012a. Multidimensional matrix characterization of equivalent double sequences. Studia Sci. Math. Hungar. 49(2): 269-281.

Patterson, R.F. & Savas, E. 2012b. Rate of P-convergence over equivalence classes of double sequence spaces. Positivity 16(4): 739-749.

Patterson, R.F. & Savas, E. 2012c. RH-conservative matrix characterization of P-convergence in probability. Comput. Math. Appl. 63(6): 1020-1025.

Patterson, R.F. & Savas, E. 2011. Matrix summability of statistically P-convergence sequences. Filomat 25(4): 55-62.

Patterson, R.F. & Savas, E. 2010a. Consistent classes of double summability methods. Appl. Math. Lett. 23(8): 831-835.

Patterson, R.F. & Savas, E. 2010b. P-asymptotically equivalent in probability. Sarajevo J. Math. 6(19): 217-228.

Patterson, R.F. & Savas, E. 2008. Summability of double independent random variables. J. Inequal. Appl. 2008: Article ID 948195.

Pringsheim, A. 1900. Zur theorie der zweifach unendlichen zahlenfolgen. Mathematische Annalen 53: 289-321.

Pruitt, W.E. 1966. Summability of independent random variables. Intern. J. Math. & Math. Sci. 15(5): 769-776.

Robison, G.M. 1926. Divergent double sequences and series. Amer. Math. Soc. Trans. 2: 50-73.

Wang, X.C. & Rao, M.B. 1985. A note on convergence of weighted sums of random variables. Internat. J Math. & Math. Sci. 8(4): 805-812.

 

 

*Pengarang untuk surat-menyurat; email: ekremsavas@yahoo.com

 

 

 

sebelumnya