Complete convergence for arrays

On the rate of complete convergence for weighted sums of arrays of Banach space valued random elements

“…No geometric conditions were imposed on the underlying Ba nach space. The Hu-Rosalsky-Szynal-Volodin [9] result unifies and extends previously obtained results in the literature in that many of them (for ex ample, results of Hsu and Robbins [7], Hu, Moricz, and Taylor [8], Gut [5], Wang, Bhaskara Rao, and Yang [18], Kuczmaszewska and Szynal [12], and Sung [16]) follow from it. Theorem 1.3 (Hu, Rosalsky, Szynal, and Volodin [9]).…”

“…, Erdos [4] This result has been generalized and extended in several directions (see Pruitt [14], Rohatgi [15], Hu, Moricz, and Taylor [8], Gut [5], Wang, Bhaskara Rao, and Yang [18], Kuczmaszewska and Szynal [12], Sung [16], and Hu, Rosalsky, Szynal, and Volodin [9] among others). Some of these articles concern a Banach space setting.…”

On Convergence of Weighted Sums of LNQD Random

“…random variables converges completely to the expected value if the variance of the summands is finite. This result has been generalized and extended in several directions and carefully studied by many authors (see, Pruitt, 1966;Rohatgi, 1971;Gut, 1992;Wang et al, 1993;Kuczmaszewska and Szynal, 1994;Magda and Sergey, 1997;Ghosal and Chandra, 1998;Hu et al, 1999Hu et al, , 2001Antonini et al, 2001;Ahmed et al, 2002;Liang et al, 2004;Baek et al, 2005). Antonini et al (2001) obtained result of the following theorem on complete and they had established some results for independent and identically distributed random variables.…”

Complete convergence for weighted sums of pairwise independent random variables

“…This dominated condition means weakly dominated, where weak refers to the fact that domination is distributional. In [27], Gut introduced a weakly mean dominated condition. We say that the random variables fX n ; n 1g are weakly mean dominated by the random variable X , where X is possibly defined on a different space if for some C > 0,…”

“…It is clear that if X dominates the sequence fX n ; n 1g in the weakly dominated sense, then it also dominates the sequence in the weakly mean dominated sense. Furthermore, Gut [27] gave an example to show that the condition (3) is weaker than the above condition (2).…”