Sains Malaysiana 45(7)(2016): 1177–1181

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

Received: 6 October 2015/Accepted: 19 December 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

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*Corresponding author; email: ekremsavas@yahoo.com