Sufficient and Necessary Conditions of Complete Convergence for Weighted Sums of ρ-Mixing Random Variables

Definition 1. A sequence of random variables is said to be a ρ-mixing sequence of random variables if there exists k ∈ N such that ρ(k) < 1. Note that if {X n , n ≥ 1} is a sequence of independent random variables, then ρ(n) = 0 for all n ≥ 1. ρ-mixing is similar to ρ-mixing, but both are quite different. ρ(k) is defined by (2) with index sets restricted to subsets S of [1, n] and subsets of T of [n + k, ∞), n, k ∈ N. On the other hand, ρ-mixing sequence assumes the condition ρ(k) → 0, but ρ-mixing sequence assumes the condition that there exists k ∈ N such that ρ(k) < 1; from this point of view, ρ-mixing is weaker than ρ-mixing.

Note that if {  ,  ≥ 1} is a sequence of independent random variables, then ρ() = 0 for all  ≥ 1. ρ-mixing is similar to -mixing, but both are quite different.() is defined by (2) with index sets restricted to subsets  of [1, 𝑛] and subsets of  of [ + , ∞), ,  ∈ .On the other hand, -mixing sequence assumes the condition () → 0, but ρ-mixing sequence assumes the condition that there exists  ∈  such that () < 1; from this point of view, ρ-mixing is weaker than -mixing.
A sequence {  ,  ≥ 1} of random variables converges completely to the constant  if In view of the Borel-Cantelli lemma, this implies that   →  almost surely.Hence, complete convergence is one of the most important problems in probability theory.Since the concept of complete convergence was introduced by Hsu and Robbins [17], there have been many authors who 2 Journal of Applied Mathematics devoted the study to complete convergence for independent and identically distributed random variables.One of the most important results is Baum and Katz theorem [18].The theorem was further generalized and extended in different ways.Katz [19] and Chow [20] formed the following generalization with a normalization of Marcinkiewicz-Zygmund type theorem for the strong law of large numbers.
Theorem 2 (see [21]).Let  ≥ 1,  > 1/2, and let {  ,  ≥ 1} be a sequence of independent and identically distributed random variables.If  ≥ 1, assume that  1 = 0. Then the following statements are equivalent: In many stochastic models, the assumption of independence among random variables is not plausible.So it is necessary to extend the concept of independence to dependence cases.Peligrad and Gut [3] extended this result from independent and identically distributed case to the case of ρ-mixing random variables with identical distribution.But they did not prove whether the result of Theorem 2 of the case  = 1 holds for ρ-mixing sequence.In practical applications it is difficult to check the independence of a sample or the samples are not independent observations.Therefore, in recent investigations limit theorems are very often considered for sequences of dependent random variables.Recently, a number of limit theorems for dependent random variables have been established by many authors.We can refer to Sung [22], Wu and Jiang [23], Wu [24], and Shen [25].
Let {  ,  ≥ 1} be a sequence of identically distributed random variables and let {  , 1 ≤  ≤ ,  ≥ 1} be an array of constants.The strong convergence results for weighted sums ∑  =1     have been studied by many authors; see, for example, Cuzick [26], Choi and Sung [27], Bai and Cheng [28], Chen and Gan [29], and so forth.Many useful linear statistics are weighted sums.Examples include least squares estimators, nonparametric regression function estimators, and jackknife estimates.
Inspired by Theorem 2.1 of Kuczmaszewska [30], our main purpose in this work is to extend the complete convergence for weighted sums ∑  =1     of independent and identically distributed random variables to the case of ρmixing random variables.However, our proven methods are different from the ones by Kuczmaszewska [30]; by applying inequality (13) of Lemma 10 our proof is much simpler than the one by Kuczmaszewska.Our proof of necessary condition (using Lemma 10) is original.We provide sufficient and necessary conditions of complete convergence for weighted sums of ρ-mixing random variables with different distributions.As applications, the Baum and Katz type result and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of ρ-mixing random variables are obtained.In addition, our main results extend and improve the corresponding results of Peligrad and Gut [3].
Throughout this paper, the symbol  denotes a positive constant which is not necessarily the same one in each appearance,   = (  ) will mean   ≤ (  ) for sufficiently large ,   ≪   will mean   = (  ), and () is the indicator function of event .

Main Results
Now we state our main results of this paper.The proofs will be given in Section 3. Theorem 3. Let  be a random variable and let {  ,  ≥ 1} be a sequence of ρ-mixing random variables satisfying the condition for all  > 0, all  ≥ 1, and some positive constant .Let {  , 1 ≤  ≤ ,  ≥ 1} be a sequence of real numbers such that where  ≍  means  = () and  = ().Let  ≥ 1,  > 1/2, and if  ≤ 1, assume that   = 0,  ≥ 1.Then the following statements are equivalent: Remark 4. When proving the limit theorem of ρ-mixing random variables with different distributions, many authors apply the condition of {  ,  ≥ 1} being stochastically dominated by , that is, for some constant  > 0, (|  | ≥ ) ≤ (|| ≥ ), for all  ≥ 0,  ≥ 1, which implies that , but the converse is not true.Hence our condition of ( 4) is weaker than the condition of stochastic dominance.
Corollary 5. Let {  ,  ≥ 1} be a sequence of ρ-mixing identically distributed random variables.Let  ≥ 1,  > 1/2, and if  ≤ 1, assume that   = 0,  ≥ 1.Then the following statements are equivalent: Remark 6. Corollary 5 not only generalizes Theorem 2 to ρmixing case, but also extends Theorem 2 of Peligrad and Gut [3] to the case  = 1.Therefore, Corollary 5 improves and extends the well-known Baum and Katz theorem.because the theorem is based on Theorem 1 [10].However, the author thinks that their proofs of Theorem 1 have a little problem, since condition (1.2) does not hold for the array with {  , 1 ≤  ≤ }.An and Yuan [10, Theorem 1] proved the implication (i) ⇒ (ii) under condition (1.3) and proved the converse under conditions (1.2) and (1.3).However, the array satisfying both (1.2) and (1.3) does not exist.Noting that ♯  /( + 1) ≤ ∑  =1 |  |  ≤ (  ), we have that  −1/ ≤ ♯  ≤ ( + 1)(  ).But this does not hold when  is fixed and  is large enough.In this paper, we obtain a new complete convergence result for weighted sums of ρmixing random variables without assumption of identical distribution.Our result generalizes and sharpens the result of An and Yuan [10].The following corollary provides the Marcinkiewicz-Zygmund type strong law of large numbers of ρ-mixing random variables without assumption of identical distribution.

Proof of Main Results
The following lemmas are useful for the proof of the main results.
Consequently, we prove our main results.
For fixed  ≥ 1, denote that If  > 1,  ≥ 1, by ( 5), (11) Note that, if  ≥ 1, we have Hence, by Kronecker lemma and ( 23 From ( 21), (22), and ( 25) we can get (20) immediately.Hence, for all  sufficiently large and any  > 0, we have max It is easy to check that for all  sufficiently large which implies that for all  sufficiently large  (max Therefore, in order to prove (ii), we only need to prove that By ( 4), (5), and  ≥ 1, we can get that That is, (29) holds.Thus, it remains to prove (30).