Complete Moment Convergence for Sung’s Type Weighted Sums of B-Valued Random Elements

Let {X n , n ≥ 1} be a sequence of random variables (or random elements) and let {a ni , 1 ≤ i ≤ n, n ≥ 1} be an array of real numbers. The weighted sums ∑n i=1 a ni X i include many useful linear statistical estimators, such as least squares estimators, nonparametric regression function estimators, and jackknife estimators. So it is interesting and meaningful to study the limiting behavior for them. In fact, many authors have studied some limiting properties. We refer to Bai and Cheng [1], Chen et al. [2], Cuzick [3], Sung [4, 5], Wang et al. [6], Wu [7], and Zhang [8]. Recently, Sung [5] obtained a complete convergence result for weighted sums of identically distributed ρ-mixing random variables (we call them Sung’s type weighted sums).


Introduction
Let {  ,  ≥ 1} be a sequence of random variables (or random elements) and let {  , 1 ≤  ≤ ,  ≥ 1} be an array of real numbers.The weighted sums ∑  =1     include many useful linear statistical estimators, such as least squares estimators, nonparametric regression function estimators, and jackknife estimators.So it is interesting and meaningful to study the limiting behavior for them.In fact, many authors have studied some limiting properties.We refer to Bai and Cheng [1], Chen et al. [2], Cuzick [3], Sung [4,5], Wang et al. [6], Wu [7], and Zhang [8].
Recently, Sung [5] obtained a complete convergence result for weighted sums of identically distributed  * -mixing random variables (we call them Sung's type weighted sums).
The weights satisfying (1) are very general.For example, set   = 1 for all 1 ≤  ≤  and  ≥ 1.Then, (1) holds for any  > 0 and therefore the weighted sums include the partial sums.Set   = 1 if 1 ≤  ≤  − 1 and   =  1/ for some  > 0.Then, (1) holds; meanwhile, (1) does not hold for any   > , and obviously the weights are unbounded in this case.So Sung's type weights are very rich and interesting, but very few authors continue to study the kind of weighted sums except Zhang [8] who obtained Theorem A for END random variables.
Chow [9] first investigated the complete moment convergence as follows.
Theorem B. Let {,   ,  ≥ 1} be a sequence of independent and identically distributed random variables with  = 0 and where  + means max{0, } for any real number .
Chen and Wang [10] pointed out that (3) is equivalent to or Li and Sp ȃ taru [11] called (4) the refined result of complete convergence.For some applications in the theory of branching processes, Sp ȃ taru [12] obtained (4) for the special case  = 2 and  = 1.
The purpose of this paper is to extend Theorem A to complete moment convergence for independent and identically distributed random elements taking values in a Banach space .We also consider the case  = 1.No geometric conditions are imposed on the Banach space.Our results also partially extend the results of Chen [16] and Li and Sp ȃ taru [20] from the partial sums to the weighted sums.

Preliminaries
Let  be a real separable Banach space with norm ‖ ⋅ ‖ and let (Ω, F, ) be a probability space.A random element  taking values in  is defined as a Borel measurable function from (Ω, F) into  with the Borel sigma-algebra.The expected value of a -valued random element  is defined as the Bochner integral and denoted by .Ledoux and Talagrand [26]).
The following assertion gives us a useful contraction principle and can be found in Lemma 6.5 of Ledoux and Talagrand [26].
The following moment inequality is due to de Acosta [27].
Lemma 4. For every  ≥ 1, there exists a positive constant   such that, for any separable Banach space  and any finite sequence {  , 1 ≤  ≤ } of independent -valued random elements with ‖  ‖  < ∞ for every 1 ≤  ≤ , the following inequalities hold: In the following,  will be used to denote various positive constants whose exact value is immaterial.
Proof.By the same argument as in the proof of Theorem 5, we can assume that  is symmetric.By the Hölder inequality, we can also assume that sup ≥1  −1 ∑  =1 |  |  ≤ 1 for 1/ <  < 2. By Proposition 1.1 in Chen and Wang [10], (37) is equivalent to Hence, it is enough to prove that If  = 1, then we take  such that V <  < 1.Then, we have by the Markov inequality, the   -inequality, the Hölder inequality, and a standard computation Therefore, (41) holds.The rest of the proof is similar to that of Theorem 5.So we complete the proof.
Remark 7. The condition  >  cannot be weakened to  > 0.
Remark 8.In the proof of Theorem 5, our method uses not only the truncation of random elements but also the truncation of weights.But in the proof of Theorem 6 we only truncate the random elements.Since the proof of Lemma 2 depends on the condition  > 1, the method of the proof of Theorem 5 cannot be applied to that of Theorem 6.If we only truncate the random elements in the proof of Theorem 5, then it is hard to estimate  * 22 when  ≥ 2 and  is not large enough.So we need two different methods to prove Theorems 5 and 6.