Acta Mathematica Academiae Scientiarum Hungaricae 65. (1994)
1994 / 1. szám - Móricz F. - Su, Kuo-Liang - Taylor, R. L.: Strong laws of large numbers for arrays of orthogonal random elements in Banach spaces
2 F MÓRICZ, KUO-LIANG SU and R L TAYLOR For quasi-orthogonal real-valued random variables {Xij}, Móricz [2] showed that the condition However, the sense of orthogonality in a Banach space must be quite different from that of the real numbers or even for Hilbert spaces. James type orthogonality for a Banach space is adopted in this paper since it is a generalized sense of orthogonality and will be described in detail in Section 2. Howell and Warren [7] proposed the sufficient condition for the one-dimensional average £ 0 where {Хг} is a sequence of 71 ;=i jB-valued random variables, В is a Ga-space, and {X,} is mutually James type orthogonal with F||X;||1+“ < oo for all t^l. However, a G^-space is a special type p space, and type p spaces will be addressed in this manuscript. Howell and Taylor [1] obtained the con ver-П gence in probability of ^2 ani^i f°r random elements in a separable Banach i = l oo oo p/ -y2 \ 51 Л [loS2(* + 1)] 2 [log2(j + 1)] 2 < OO i=l j = 1 J implies ^ m n lim ----} Xi1 = 0 a.s. max{m,n}—»-oo 771 Tl ~J r—f г=1 j = 1 He also proved that oo oo 2 H 'Epp [l0§2(i + 1)] 2 [log2(j + 1)] 2 < OO i=l j=l J is, in certain particular cases, the necessary condition for ^ in a lim ----У У Xij = 0 a.s. m,n—* oo mn •=1 j=l E E|Wl‘ + г1+°' log1+aг < oo, 0 < a ^ 1, Acta Mathemaiica Hungarica 65, 1994