Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 948195, 12 pages doi:10.1155/2008/948195 Research Article Summability of Double Independent Random Variables Richard F. Patterson 1 and Ekrem Savas¸ 2 1 Department of Mathematics and Statistics, University of North Florida, 1 UNF Drive, Jacksonville, FL 32224, USA 2 Department of Mathematics, Istanbul commerce University, Uskudar, 34672 Istanbul, Turkey Correspondence should be addressed to Richard F. Patterson, rpatters@unf.edu Received 21 May 2008; Accepted 1 July 2008 Recommended by Jewgeni Dshalalow We will examine double sequence to double sequence transformation of independent identically distribution random variables with respect to four-dimensional summability matrix methods. The main goal of this paper is the presentation of the following theorem. If max k,l |a m,n,k,l |  max k,l |a m,k a n,l |  Om −γ 1 On −γ 2 , γ 1 ,γ 2 > 0, then E| ˘ X| 11/γ 1 < ∞ and E| ˘ ˘ X| 11/γ 2 < ∞ imply that Y m,n → μ almost sure P-convergence. Copyright q 2008 R. F. Patterson and E. Savas¸. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let X k,l  be a factorable double sequence of independent, identically distributed random variables with E|X k,l | < ∞ and EX k,l μ.LetA  a m,n,k,l be a factorable double sequence to double sequence transformation defined as Ax m,n  ∞,∞  k,l1,1 a m,n,k,l x k,l . 1.1 These factorable sequences and matrices will be used to characterize such transformations with respect to Robison and Hamilton-type conditions see 1, 2. That is,regularity conditions of the following type. The four-dimensional matrix A is RH-regular if and only if RH 1 : P-lim m,n a m,n,k,l  0 for each k and l; RH 2 : P-lim m,n  k,l a m,n,k,l  1; RH 3 : P-lim m,n  k |a m,n,k,l |  0 for each l; RH 4 : P-lim m,n  l |a m,n,k,l |  0 for each k; RH 5 :  k,l |a m,n,k,l | is P-convergent; and RH 6 : there exist positive numbers A and B such that  k,l>B |a m,n,k,l | <A. 2 Journal of Inequalities and Applications Throughout this paper, we will denote  ∞,∞ k,l1,1 a m,n,k,l X k,l by Y m,n and examine Y m,n with respect to the Pringsheim converges. To accomplish this goal, we begin by presenting and prove the following theorem. A necessary and sufficient condition that Y m,n  ˘ Y m ˘ ˘ Y n P-converges to μ in probability is that max k,l |a m,n,k,l |  max k,l |a m,k a n,l | converges to 0 in the Pringsheim sense. This theorem and other similar to it will be used in the pursuit of establishing the following. If max k,l |a m,n,k,l |  max k,l |a m,k a n,l |  Om −γ 1 On −γ 2 , γ 1 ,γ 2 > 0, then E| ˘ X| 11/γ 1 < ∞,E| ˘ ˘ X| 11/γ 2 < ∞ 1.2 implies that Y m,n → μ almost sure P-convergence. 2. Definitions, notations, and preliminary results Let us begin by presenting Pringsheim’s notions of convergence and divergence of double sequences. Definition 2.1 see 3. A double sequence x x k,l  has Pringsheim limit L denoted by P- lim x  L provided that given >0 there exists N ∈ N such that |x k,l − L| <whenever k, l > N. We will describe such an x more briefly as “P-convergent.” Definition 2.2. A double sequence x is called definite divergent, if for every arbitrarily large G>0 there exist two natural numbers n 1 and n 2 such that |x n,k | >Gfor n ≥ n 1 ,k≥ n 2 . Throughout this paper, we will also denote  ∞,∞ k,l1,1 by  k,l . Using these definitions, Robison and Hamilton presented a series of concepts and matrix characterization of P-convergence. The first definition they both presented was the following. The four- dimensional matrix A is said to be RH-regular if it maps every bounded P-convergent sequence into a P-convergent sequence with the same P-limit. The assumption of bounded- ness was made because a double sequence which is P-convergent is not necessarily bounded. They both independently presented the following Silverman-Toeplitz type characterization of RH-regularity 4, 5. Theorem 2.3. The four-dimensional matrix A is RH-regular if and only if RH 1 : P-lim m,n a m,n,k,l  0 for each k and l; RH 2 : P-lim m,n  k,l a m,n,k,l  1; RH 3 : P-lim m,n  k |a m,n,k,l |  0 for each l; RH 4 : P-lim m,n  l |a m,n,k,l |  0 for each k; RH 5 :  k,l |a m,n,k,l | is P-convergent; and RH 6 : there exist positive numbers A and B such that  k,l>B |a m,n,k,l | <A. Following Robison and Hamilton work, Patterson in 6 presented the following two notions of subsequence of a double sequence. Definition 2.4. The double sequence y is a double subsequence of the sequence x provided that there exist two increasing double index sequences {n j } and {k j } such that if z j  x n j ,k j , R. F. Patterson and E. Savas¸3 then y is formed by z 1 z 2 z 5 z 10 z 4 z 3 z 6 — z 9 z 8 z 7 — ————. 2.1 Definition 2.5 Patterson 6. A number β is called a Pringsheim limit point of the double sequence x provided that there exists a subsequence y of x that has Pringsheim limit β: P-limyβ. Using these definitions, Patterson presented a series of four-dimensional matrix characterizations of such sequence spaces. Let {x k,l } be a double sequence of real numbers and, for each n,letα n  sup n {x k,l : k, l ≥ n}. Patterson 7 also extended the above notions with the presentation of the following. The Pringsheim limit superior of x is defined as follows: 1 if α ∞ for each n, then P-lim sup x :∞; 2 if α<∞ for some n, then P-lim sup x : inf n {α n }. Similarly, let β n  inf n {x k,l : k, l ≥ n}. Then the Pringsheim limit inferior of x is defined as follows: 1 if β n  −∞ for each n, then P-lim inf x : −∞; 2 if β n > −∞ for some n, then P-lim inf x : sup n {β n }. 3. Main result The analysis of double sequences of random variables via four-dimensional matrix transformations begins with the following theorem. However, it should be noted that the relationship between our main t heorem that is stated above and the next four theorems will be apparent in their statements and proofs. Theorem 3.1. A necessary and sufficient condition that Y m,n  ˘ Y m ˘ ˘ Y n P-converges to μ in probability is that max k,l |a m,n,k,l |  max k,l |a m,k a n,l | converges to 0 in the Pringsheim sense. Proof. First, note that lim ˘ T→∞ ˘ TP| ˘ X|≥ ˘ T0, lim ˘ ˘ T→∞ ˘ ˘ TP| ˘ ˘ X|≥ ˘ ˘ T0 3.1 because E| ˘ X| < ∞ and E| ˘ ˘ X| < ∞.LetT  ˘ T ˘ ˘ T, X m,n,k,l  ˘ X m,k ˘ ˘ X n,l , a m,n,k,l X k,l  a m,k ˘ X k a n,l ˘ ˘ X l , and Z m,n  ˘ Z m ˘ ˘ Z n   k,l X m,n,k,l . For sufficiently large m and n and since max k,l |a m,n,k,l | is 4 Journal of Inequalities and Applications a P-null sequence, it follows from 3.1 that PZ m,n /  Y m,n  ≤  k,l PX m,n,k,l /  a m,n,k,l X k,l    k,l P  | ˘ X|≥ 1 |a m,k | ; | ˘ ˘ X|≥ 1 |a n,l |  ≤   k,l |a m,n,k,l | ≤ M, 3.2 where M is define by RH 6 of regularity conditions. Therefore, it suffices to show that P-lim m,n Z m,n  μ in probability. 3.3 Observe that EZ m,n  − μ   k,l a m,n,k,l   | ˘x|<1/|a m,k | ˘xd ˘ F  | ˘ ˘x|<1/|a n,l | ˘ ˘xd ˘ ˘ F − μ   μ   k,l a m,n,k,l − 1  , 3.4 which is a P-null sequence. Since 1 ˘ T ˘ ˘ T  | ˘x|< ˘ T  | ˘ ˘x|< ˘ ˘ T ˘x 2 ˘ ˘x 2 d ˘ Fd ˘ ˘ F  1 ˘ T ˘ ˘ T {− ˘ T 2 P| ˘ X|≥ ˘ T · − ˘ ˘ T 2 P| ˘ ˘ X|≥ ˘ ˘ T}  1 ˘ T ˘ ˘ T  2  ˘ T 0 ˘xP| ˘ X|≥ ˘xd ˘x · 2  ˘ ˘ T 0 ˘ ˘xP| ˘ ˘ X|≥ ˘ ˘xd ˘ ˘x  3.5 is a P-null sequence with respect to T, we have  k,l Var X m,n,k,l ≤  k,l |a m,n,k,l | 2  | ˘x|<1/|a m,k | ˘x 2 d ˘ F  | ˘ ˘x|<1/|a n,l | ˘ ˘x 2 d ˘ ˘ F ≤   k,l |a m,n,k,l |≤M 3.6 for m and n sufficiently large, where F  ˘ F ˘ ˘ F and x  ˘x ˘ ˘x.ItisalsoclearthatE  k,l x m,n,k,l  2 is finite. Thus,  k,l Var X m,n,k,l  Var   k,l X m,n,k,l  3.7 is finite. The result clearly follows from the Chebyshev’s inequality. Thus, the sufficiency is proved. Now, let us consider the necessary part of this theorem. Similar to Pruitt’s notation 8, let U k,l  X k,l − μ and consider the transformation T m,n   k,l a m,n,k,l U k,l . Our goal become showing that T m,n P-converges in probability to 0. Which imply that T m,n P-converges in law to 0. Let us consider the characteristic function of T m,n, that is, Ee uT m,n Ee u  k,l a m,n,k,l U k,l EΠ k,l e ua m,n,k,l U k,l Π k,l Ee ua m,n,k,l U k,l  :Π k,l gua m,n,k,l . 3.8 R. F. Patterson and E. Savas¸5 Observe that P-lim m,n {Π k,l gua m,n,k,l }  1. 3.9 Because |Π k,l gua m,n,k,l |≤|gua m,n,k,l |≤1 3.10 for all m, n we have that P-lim m,n gua m,n,k,l 1 3.11 for all k,l. Clearly, there exists u 0 such that |gua m,n,k,l | < 1for0< |u| <u 0 .Letu  u 0 /2M then there exists a double subsequence a m,n,k m ,l n  such that |ua m,n,k m ,l n |≤Mu  u 0 2 . 3.12 Thus P-lim m,n ua m,n,k m ,l n  0. Therefore, clearly we can choose k m ,l n  such that |a m,n,k m ,l n |  max k,l |a m,n,k,l |. 3.13 Theorem 3.2. If E| ˘ X| 11/γ 1 < ∞, E| ˘ ˘ X| 11/γ 2 < ∞, and max k,l |a m,n,k,l |  max k |a m,k |· max l |a n,l |≤ ˘ Bm −γ 1 ˘ ˘ Bn −γ 2 , then for every >0  m,n P|a m,n,k,l X k,l |≥ for some k, l < ∞, 3.14 that is,  m,n P|a m,k ˘ X k |≥; |a n,l ˘ ˘ X l |≥ for some k, l < ∞. 3.15 Proof. Let N m,n xN m,n  ˘x ˘ ˘x  {k,l:1/|a m,k |≤ ˘x;1/|a n,l |≤ ˘ ˘x} |a m,n,k,l |. 3.16 Note x  ˘x ˘ ˘x, and observe that N m,n x0for ˘x<m γ 1 , ˘ ˘x<n γ 2 , and  ∞ 0 dN m,n x   k,l |a m,n,k,l |≤M.If GxP|X|≥xP| ˘ X|≥ ˘xP| ˘ ˘ X|≥ ˘ ˘xG ˘xG ˘ ˘x, 3.17 6 Journal of Inequalities and Applications then xGx converges to 0 in the Pringsheim sense because EX < ∞ and recalled t hat T  ˘ T ˘ ˘ T. Therefore,  k,l P|a m,n,k,l x k,l |≥1  k,l G  1 |a m,n,k,l |    k,l 1 |a m,n,k,l | G  1 |a m,n,k,l |  |a m,n,k,l |   ∞ 0 xGxdN m,n x  N m,n TTGT| ∞ 0 | ∞ 0 −  ∞ 0 N m,n xdxGx  lim T→∞ N m,n TTGT −  ∞ 0 N m,n xdxGx ≤ M  ∞ m γ 1  ∞ n γ 2 |dxGx|  M  ∞ m γ 1  ∞ n γ 2 |d ˘xG ˘xd ˘ ˘xG ˘ ˘x|. 3.18 Our goal now is to get an estimate for  ∞ m γ 1  ∞ n γ 2 |dxGx|. To this end observe that, for z<y yGy − zGzy − zGzyGz − Gy, 3.19 where y − zGz and yGz − Gy are increasing and decreasing functions of y, respectively. Thus  ˘y ˘z  ˘ ˘y ˘ ˘z d|xGx|≤ ˘y − ˘zG ˘z ˘yG ˘z − G ˘y ·  ˘ ˘y − ˘ ˘zG ˘ ˘z ˘ ˘yG ˘ ˘z − G ˘ ˘ Y. 3.20 The last inequality grant us the following:  ∞ m γ 1  ∞ n γ 2 |d ˘xG ˘xd ˘ ˘xG ˘ ˘x|  ∞,∞  i,jm,n  i1 γ 1 i γ 1  j1 γ 2 j γ 2 |d ˘xG ˘xd ˘ ˘xG ˘ ˘x| ≤ ∞,∞  i,jm,n {i  1 γ 1 − i γ 1 Gi γ 1  · j  1 γ 2 − j γ 2 Gj γ 2 }  ∞,∞  i,jm,n {i  1 γ 1 Gi γ 1  − Gi  1 γ 1 · j  1 γ 2 Gj γ 2  − Gj  1 γ 2 }. 3.21 R. F. Patterson and E. Savas¸7 Therefore,  ∞ m γ 1  ∞ n γ 2 |d ˘xG ˘xd ˘ ˘xG ˘ ˘x| ≤ 2 ∞,∞  i,jm,n {i  1 γ 1 Gi γ 1  − Gi  1 γ 1 · j  1 γ 2 Gj γ 2  − Gj  1 γ 2 }. ∞  m,n P|a m,n,k,l X k,l |≥ for some k, l ≤ ∞  m,n ∞  k,l P|a m,n,k,l X k,l |≥ ≤ 2M ∞,∞  m,n1,1 ∞,∞  i,jm,n {i1 γ 1 Gi γ 1 −Gi1 γ 1 · j1 γ 2 Gj γ 2 −Gj 1 γ 2 }  2M ∞,∞  i,j1,1 {i  1 γ 1 Gi γ 1  − Gi  1 γ 1 · j  1 γ 2 Gj γ 2  − Gj  1 γ 2 } ≤ 2 1γ 1 2 1γ 2 M  | ˘x| 11/γ 1 | ˘ ˘x| 11/γ 2 d ˘ F ˘xd ˘ ˘ F ˘ ˘x < ∞. 3.22 Theorem 3.3. Let x and F be define as in Theorem 3.2.IfE| ˘ X| 11/γ 1 < ∞, E| ˘ ˘ X| 11/γ 2 < ∞, and max k,l |a m,n,k,l |  max k |a m,k |·max l |a n,l |≤ ˘ Bm −γ 1 ˘ ˘ Bn −γ 2 then for α 1 <γ 1 /2γ 1 1 and α 2 <γ 2 /2γ 2  1  m,n P|a m,n,k,l X k,l |≥m α 1 n α 2 for at least two pairs k, l < ∞, 3.23 that is,  m,n P|a m,k ˘ X k |≥m α 1 ; |a n,l ˘ ˘ X l |≥n α 2 for at least two pairs k, l < ∞. 3.24 Proof. By Markov’s inequality, we have the following:  m P|a m,k ˘ X k |≥m α 1  ≤|a m,k | 11/γ 1 E| ˘x| 11/γ 1 m α 1 11/γ 1  ,  n P|a n,l ˘ ˘ X l |≥n α 2  ≤|a n,l | 11/γ 2 E| ˘ ˘x| 11/γ 2 n α 2 11/γ 2  . 3.25 8 Journal of Inequalities and Applications Therefore,  m,n P|a m,k ˘ X k |≥m α 1 ; |a n,l ˘ ˘ X l |≥n α 2 for at least two pairs k, l ≤  i /  k,j /  l P|a m,i ˘ X i |≥m α 1 ; |a m,k ˘ X k |≥m α 1 ; |a n,j ˘ ˘ X j |≥n α 2 ; |a n,l ˘ ˘ X l |≥n α 2  ≤ E| ˘x| 11/γ 1  2 m 2α 1 11/γ 1   i /  k |a m,i | 11/γ 1 |a m,k | 11/γ 1 · E| ˘ ˘x| 11/γ 2  2 n 2α 2 11/γ 2   j /  l |a n,j | 11/γ 2 |a n,l | 11/γ 2 ≤ E| ˘x| 11/γ 1  2 · E| ˘ ˘x| 11/γ 2  2 ˘ B 2/γ 1 ˘ ˘ B 2/γ 2 M 4 m 2−1α 1 11/γ 1  n 2−1α 2 11/γ 2  , 3.26 which is P-convergent when sum on n and m provided that α 1 <γ 1 /2γ 1  1 and α 2 < γ 2 /2γ 2  1. Theorem 3.4. Let x and F be define as in Theorem 3.2.Ifμ  0, E| ˘ X| 11/γ 1 < ∞, E| ˘ ˘ X| 11/γ 2 < ∞, and max k,l |a m,n,k,l |  max k |a m,k |·max l |a n,l |≤ ˘ Bm −γ 1 ˘ ˘ Bn −γ 2 then for >0  m,n P    k,l |a m,n,k,l X k,l |≥  < ∞, 3.27 where   k,l a m,n,k,l X k,l   {k:|a m,k X k |<m −α 1 l:|a n,l X l |<n −α 2 } a m,n,k,l X k,l , 3.28 α 1 <γ 1 , and α 2 <γ 2 . Proof. Let X m,n,k,l : ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ X m,k ;if|a m,k X k | <m −α 1 , X n,l ;if|a n,l X l | <n −α 2 , 0; otherwise, 3.29 and β m,n,k,l  EX m,n,k,l .Ifa m,n,k,l  0, then β m,n,k,l  μ  0andifa m,n,k,l /  0, then |β m,n,k,l |      μ −  | ˘x|≥m −α 1 |a m,k | −1  | ˘ ˘x|≥m −α 2 |a n,l | −1 xdF     ≤  | ˘x|≥m −α 1 ˘ B −1 m γ 1  | ˘ ˘x|≥n −α 2 ˘ ˘ B −1 n γ 2 |x|dF. 3.30 Therefore, P-lim m,n β m,n,k,l  0 uniformly in k,l and P-lim m,n  k,l a m,n,k,l β m,n,k,l  0. Let Z m,n,k,l  Z m,k Z n,l  X m,n,k,l − β m,n,k,l , 3.31 R. F. Patterson and E. Savas¸9 so that EZ m,n,k,l 0, E|Z m,k | 11/γ 1  <c 1 , and E|Z n,l | 11/γ 2  <c 2 for some c 1 and c 2 .Also |a m,k Z m,k |≤2m −α 1 and |a n,l Z n,l |≤2n −α 2 . Observe that   k,l a m,n,k,l X k,l   k,l a m,n,k,l X m,n,k,l   k,l a m,n,k,l Z m,n,k,l   k,l a m,n,k,l β m,n,k,l . 3.32 Note for sufficiently large m and n         k,l a m,n,k,l X k,l      ≥   ⊂        k,l a m,n,k,l Z m,n,k,l      ≥  2  . 3.33 Thus it is sufficient to show that  m,n P         k,l |a m,n,k,l Z m,n,k,l |      ≥   < ∞. 3.34 Let η 1 and η 2 be the least integers greater than 1/γ 1 and 1/γ 2 , respectively. Our goal now is to produce an estimate for E   k a m,k Z m,k  2η 1   l a n,l Z n,l  2η 2  . 3.35 Observe that E   k a m,k Z m,k  2η 1   l a n,l Z n,l  2η 2  3.36 is equal to  k 1 ,k 2 , ,k 2p ; l 1 ,l 2 , ,l 2q E  2p  i1 2q  j1 a m,n,k i ,l j Z m,n,k i ,l j  . 3.37 It happens to be the case that E  k a m,k Z m,k  2η 1   l a n,l Z n,l  2η 2  is zero if k i ,l i /  k j ,l j for i /  j because the Z m,n,k,l ’s are independent and EZ m,n,k,l 0. Let us now consider the general term. Thus p 1 of the k  s  φ 1 , ,p θ 1 of the k  s  φ θ 1 , q 1 of the k  s  ϕ 1 , ,q θ 2 of the k  s  ϕ θ 2 , r 1 of the l  s  κ 1 , ,r τ 1 of the l  s  κ τ 1 , s 1 of the l  s  ω 1 , ,s τ 2 of the l  s  ω τ 2 , 3.38 where 2 ≤ p i ≤ 1  1/γ 1 , q j > 1  1/γ 1 ,2≤ r λ ≤ 1  1/γ 2 , s χ > 1  1/γ 2 , θ 1  i1 p i  θ 2  j1 q j  2η 1 , τ 1  λ1 r i  τ 2  χ1 s χ  2η 2 . 3.39 10 Journal of Inequalities and Applications Now let us consider the following expectation: E  θ 1  i1 a m,φ i Z m,φ i  p i · θ 2  j1 a m,ϕ j Z m,ϕ j  q j · τ 1  λ1 a n,κ λ Z n,κ λ  r λ τ 2  χ1 a n,ω χ Z n,ω χ  s χ  ≤ 1  c 1  θ 1 1  c 2  τ 1 · τ 2  χ1 |a m,φ i | p i τ 1  λ1 |a n,κ λ | r λ · E  θ 2  j1 a m,ϕ j Z m,ϕ j  q j · τ 2  χ1 a n,ω χ Z n,ω χ  s χ  ≤ 1  c 1  θ 1 1  c 2  τ 1 · θ 1  i1 |a m,φ i | p i τ 1  λ1 |a n,κ λ | r λ · θ 2  j1 |a m,ϕ j | 11/γ 1 2m −α 1  q j −1−1/γ 1 · τ 2  χ1 |a n,ω χ | 11/γ 2 2n −α 2  s χ −1−1/γ 2 ≤ 1  c 1  θ 1 1  c 2  τ 1 · θ 1  i1 |a m,φ i ||a m,φ i | p i −1 · τ 1  λ1 |a n,κ λ ||a n,κ λ | r λ −1 · θ 2  j1 |a m,ϕ j | 11/γ 1 2m −α 1  q j −1−1/γ 1 · τ 2  χ1 |a n,ω χ | 11/γ 2 2n −α 2  s χ −1−1/γ 2 ≤ 1  c 1  θ 1 1  c 2  τ 1 · θ 1  i1 |a m,φ i | τ 1  λ1 |a n,κ λ | θ 2  j1 |a m,ϕ j | τ 2  χ1 |a n,ω χ | ·  ˘ Bm −γ 1   θ 1 i1 p i −1θ 2 /γ 1 2m −α 1   θ 2 j1 q j −1−1/γ 1  ·  ˘ ˘ Bn −γ 2   τ 1 λ1 r λ −1τ 2 /γ 2 2n −α 2   τ 2 χ1 s χ −1−1/γ 2  , 3.40 where c 1 and c 2 are upper bound for E|Z m,k | and E|Z n,l |, respectively. Now let us examine the negative exponents, that is, γ 1 θ 1  i1 p i − 1θ 2  α 1 θ 2  j1  q j − 1 − 1 γ 1  , γ 2 τ 1  λ1 r λ − 1τ 2  α 2 τ 2  χ1  s χ − 1 − 1 γ 2  . 3.41 Observe that, if θ 2 and τ 2 are 1 or large, then θ 2  α 1 θ 2  j1  q j − 1 − 1 γ 1  ≥ 1  α 1  η 1 − 1 γ 1  , τ 2  α 2 τ 2  χ1  s χ − 1 − 1 γ 2  ≥ 1  α 2  η 2 − 1 γ 2  , 3.42 [...]... Patterson, “Analogues of some fundamental theorems of summability theory,” International Journal of Mathematics and Mathematical Sciences, vol 23, no 1, pp 1–9, 2000 7 R F Patterson, Double sequence core theorems,” International Journal of Mathematics and Mathematical Sciences, vol 22, no 4, pp 785–793, 1999 8 W E Pruitt, Summability of independent random variables, ” Journal of Mathematics and Mechanics,... This research was completed with the support of The Scientific and Technological Research Council of Turkey while the first author was a visiting scholar at Istanbul Commerce University, Istanbul, Turkey 12 Journal of Inequalities and Applications References 1 H J Hamilton, “Transformations of multiple sequences,” Duke Mathematical Journal, vol 2, no 1, pp 29–60, 1936 2 G M Robison, “Divergent double. .. P-convergence 1/γ1 < m,n Proof Observe that am,n,k,l Xk,l − μ am,n,k,l Xk,l k,l k,l μ am,n,k,l 3.46 k,l Note the last term P-converge to μ because of the regularity of A We will only consider the case when μ 0 By the Borel-Cantelli lemma, it is sufficient to prove that for > 0 am,n,k,l xk,l ≥ P m,n ≤ ∞ 3.47 k,l At this point, the proof follows a path identical to Pruitt’s proof using the above theorems... Robison, “Divergent double sequences and series,” Transactions of the American Mathematical Society, vol 28, no 1, pp 50–73, 1926 3 A Pringsheim, “Zur theorie der zweifach unendlichen Zahlenfolgen,” Mathematische Annalen, vol 53, no 3, pp 289–321, 1900 4 L L Silverman, On the definition of the sum of a divergent series, Ph.D thesis, University of Missouri Studies, Mathematics Series, Columbia, Mo, USA,... product of θ1 θ2 |am,φi | K1 i 1 τ1 |an,κλ | K2 j 1 τ2 |am,ϕj |m−1−α1 η1 −1/γ1 , 3.44 −1−α2 η2 −1/γ2 , |an,ωχ |n χ 1 λ 1 ˘ ˘ where K1 dependent on c1 , γ1 , B; and c2 , γ2 , B, respectively Therefore, 2η1 ≤ K3 m−1−α1 am,k Zm,k E η2 −1/γ1 , k 3.45 2η2 E an,l Zn,l ≤ K4 n−1−α2 η2 −1/γ2 l ˘ ˘ for some K3 and K4 which independent on c1 , γ1 , B, M and c2 , γ2 , B, M, respectively With both independent of m, . Corporation Journal of Inequalities and Applications Volume 2008, Article ID 948195, 12 pages doi:10.1155/2008/948195 Research Article Summability of Double Independent Random Variables Richard. examine double sequence to double sequence transformation of independent identically distribution random variables with respect to four-dimensional summability matrix methods. The main goal of this. be a factorable double sequence of independent, identically distributed random variables with E|X k,l | < ∞ and EX k,l μ.LetA  a m,n,k,l be a factorable double sequence to double sequence