On Complete Convergence and Strong Law for Weighted Sums of i.i.d. Random Variables

and Applied Analysis 3 Remark 5. Let g(x) = √x logx and a nk = g −1 (n) for 1 ≤ k ≤ n and n ≥ 1. Then, byTheorem 1, the moment conditions EX = 0, EX2 = 1, and E(X2/ log |X|)r+1 < ∞ imply that


Introduction and the Main Result
Stout [1] obtained the following celebrated result.
Theorem A (see Theorem 4.1.3in [1] or p. 1556 in Stout [2]).Let {,   ≥ 1} be a sequence of independent and identically distributed random variables with  = 0 and || 2/ < ∞ for some 0 <  ≤ 1. Suppose that {  , 1 ≤  ≤ ,  ≥ 1} is a sequence of constants with Formula ( 3) is called complete convergence and this concept was introduced by Hsu and Robbins [3].Sung [4] and Cheng and Wang [5] extended Theorem A to random elements taking values in a Banach space.Wu [6] and Sung [7] extended Theorem A to negatively dependent random variables.It should be pointed out that they all used some exponential inequalities to prove their result and hence the proofs are similar to that of Theorem A except for more computational complexity.
Next, we consider the case of 0 <  ≤ 1/2.Lai [10] showed that for  > √ 2 provided  = 0,  2 = 1, and  4 /log 2 || < ∞.Hence, for any  > 0, Then (1) and ( 2) hold for  = 1/2.Note that the moment conditions are weaker than those of Theorem A when  = 1/2.Hence, the moment conditions of Theorem A may not be optimal for a special case of weighted sums.But it is not known whether the moment conditions of Theorem A are not optimal when 0 <  ≤ 1/2.
In this paper, we will discuss the optimized moment conditions of Theorem A when 0 <  ≤ 1/2.We obtain a more generalized complete convergence result for weighted sums which includes the result of Lai [10].Our method used is completely different from those in Lai [10] and Stout [1].
Li et al. [11] obtained the following celebrated result.
Suppose that {  , 1 ≤  ≤ ,  ≥ 1} is a sequence of constants with (1) for some constant 0 <  < ∞ and where Theorem B has been extended and improved by many authors.Jing and Liang [12] extended and improved to negatively associated (NA) random variables, Budsaba et al. [13] to certain types of -statistics bases on this kind of weighted sums of NA random variables, and Thanh and Yin [14] to the random weighted sums.In particular, under the condition lim sup Jing and Liang [12] showed that lim sup when 0 <  < 1/2.Is it possible to find the sharp bound of (10)?In this paper, we will give a definite answer to the question under more general case.
Remark 2. Recall that a measurable function () is said to be regularly varying at infinity with index We refer to Bingham et al. [15] for other equivalent definitions and for detailed and comprehensive study of properties of regularly varying functions.For example, if where   > 0 and   > 0 are constants depending only on .
Hence, ( 11) and ( 12) hold.Under the moment conditions of Theorem 1, for all  > √ 2.On the other hand, it is easy to show that if the above formula holds for some  > 0, then ℎ  (||)|| 2 / log ℎ(||) < ∞ by the similar argument as in Lai [10].Thus, the moment conditions of Theorem 1 are sharp in the sense that the moment conditions on  cannot be weakened.
By Theorem 1 and Borel-Cantelli lemma, we have the following corollary.Corollary 6.Under the conditions of Theorem 1, let  = 1, {  , 1 ≤  ≤ ,  ≥ 1} be an array of independent random variables with the same distribution as .Then In particular, the moment conditions  = 0,  2 = 1, and Remark 7. Formula ( 21) is called the law of single logarithm which is due to Hu and Weber [16].They proved it under the strong moment condition  4 < ∞.Qi [17] and Li et al. [18] independently proved that ( 21) is equivalent to conditions  = 0,  2 = 1, and  4 /log 2 || < ∞.In particular, Li et al. [18] gave a version of random elements taking values in a Banach space.
Remark 10.Let ,  > 0 and Then, it is easy to show that max for some  > 0 and

Proofs of the Main Results
The main idea in the proofs of the main results is to use the following invariance principle (see Sakhanenko [22][23][24]), which is a powerful tool in the field of limit theory (e.g., see Csörgő et al. [25], Jiang and Zhang [26], Chen and Gan [21], and Chen and Wang [27]).
The second series converges by  21 < ∞.But by using {|| > } ∼ √2/ −1  − 2 /2 , it is easy to show that the series on the left-hand side diverges.Hence, the last series on the righthand side also diverges.That is, (40) holds.