On the rate of complete convergence for weighted sums of arrays of Banach space valued random elements with application to moving average processes

Complete Convergence for Weighted Sums of Random Elements

“…No assumptions are given on the geometry of the underlying Banach space. The result generalizes the main results of Ahmed et al [1], Chen et al [2], and Volodin et al [14]. …”

“…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.…”

“…In particular, the case t = −1 is of interest. Ahmed et al [1] conjectured that when t = −1, the assumption E|X| ν < ∞ can be replaced by E|X| 1+µ/γ log ρ (|X|) < ∞ (ρ > 0) in Theorem 1.2. Sung and Volodin [13] gave a positive answer as follows: Theorem 1.4 (Sung and Volodin [13]).…”

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.…”

On complete convergence for arrays of rowwise dependent random variables

“…and Yang (1993), Kuczmaszewska and Szynal (1988, 1991, 1994, Sung (1997) Bozorgnia, Patterson andTaylor (1993), Hu, Rosalsky, Szynal and Volodin (1999), Hu, Szynal and Volodin (1998), Hu and Volodin (2000), Ahmed, Antonini and Volodin (2002), Kuczmaszewska (2004), Sung, Volodin and Hu (2005), T om acs (2005)). In this paper we consider a complete convergence in the strong law of large numbers for arrays of dependent random variables.…”