On complete convergence for arrays of rowwise independent random elements in banach spaces

Abstract

We extend and generalize some recent results on complete convergence (cf. Hu, Moricz, and Taylor [14], Gut [ll], Wang, Bhaskara Rao, and Yang [26], Kuczmaszewska and Szynal 1171, and Sung [23]) for arrays of rowwise independent Banach space valued random elements. In the main result, no assumptions are made concerning the existence of expected values or absolute moments of the random elements and no assumptions are made concerning the geometry of the underlying Banach space. Some well-known results from the literature are obtained easily as corollaries. The corresponding convergence rates are also established. 963

On the complete convergence of weighted sums for arrays of negatively associated variables

“…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, (see Baum and Katz (1965), Hu, Rosalsky, Szynal, and Volodin (1999), Hu, Li, Rosalsky, and Volodin (2001)), Gut (1992), Wang, Rao, and Yang (1993), Kuczmaszewska and Szynal (1994), Li, Rao, and Wang (1992), Li, Rao, Ting, and Wang (1995), Zhang (1996), Ghosal and Chandra (1998), and Ahmed, Antonini, and Volodin (2002). In particular, Hu et al (2001) obtained the following result in Banach space.…”

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 Random Elements

“…Some of these generalizations are in a Banach space setting, for example, see Ahmed et al [1], Hu et al [5,6], Kuczmaszewska and Szynal [7], Sung [10], Volodin et al [14], and Wang et al [15]. A sequence of Banach space valued random elements is said to converge completely to the 0 element of the Banach space if the corresponding sequence of norms converges completely to 0.…”

“…Hu et al [6] presented a general result establishing complete convergence for the row sums of an array of rowwise independent but not necessarily identically distributed Banach space valued random elements. Using this, Hu et al [5] obtained the following complete convergence result.…”