Baum-Katz’s Type Theorems for Pairwise Independent Random Elements in Certain Metric Spaces

In this study, some Baum-Katz’s type theorems for pairwise independent random elements are extended to a metric space endowed with a convex combination operation. Our results are considered in the cases of identically distributed and non-identically distributed random elements. Some illustrative examples are provided to sharpen the results.


Introduction
The concept of complete convergence for a sequence of random variables (r.v.'s) was introduced by Hsu and Robbins [6]. A sequence {X n , n 1} of real-valued r.v.'s converges completely to a constant θ if ∞ n=1 P (|X n − θ| > ε) < ∞ for any ε > 0, and hence it follows from Borel-Cantelli's lemma that X n → θ almost surely. Also in [6], Hsu and Robbins proved that the sequence of arithmetic means of independent and identically distributed (i.i.d.) r.v.'s converges completely to the common expected value if their variance is finite. This result has been considered and extended by many authors. A noteworthy result was obtained by Katz [7] and Baum and Katz [2], that is: Theorem 1.1 Let {X, X n , n 1} be a sequence of i.i.d. r.v.'s, and set S n = n i=1 X i . Given p 1 and 0 < r < 2. Then E|X| pr < ∞ if and only if ∞ n=1 n p−2 P (max 1 k n |S k − km| > εn 1/r ) < ∞ for every ε > 0, where m = EX if pr 1 and m = 0 if 0 < pr < 1. This result has been extensively studied for various classes of r.v.'s. Recently, Bai et al. [1] considered a particular case of above Baum-Katz's result when r = 1 and 1 p < 2, and in this situation, the condition of i.i.d. can be relaxed to be pairwise i.i.d. [1,Theorem 1.2]. Moreover, in Banach space setting, Bai et al. [1] also derived a similar result without any geometric property of the underlying Banach space [1,Theorem 3.2].
Besides considering Baum-Katz's type theorems for various classes of r.v.'s (e.g., pairwise i.i.d. r.v.'s in [1] or martingale, negatively associated r.v.'s, ρ * -mixing r.v.'s in [8]), many researchers have also extended them into more abstract spaces such as Hilbert spaces [5] or Banach spaces [1]. Continuing this direction, we will discuss Baum-Katz's type theorems in a convex combination space, which is the certain metric space introduced in 2006 by Terán and Molchanov [12]. Roughly speaking, a convex combination space is a metric space endowed with a convex combination operation and the extension from linear spaces to convex combination spaces is not trivial. Some very basic sets, such as singletons and balls, may fail to be convex in this type of metric spaces. To illustrate this demonstration, Terán and Molchanov [12] provided many interesting examples for convex combination spaces, for example, the space of all cumulative distribution functions and the space of upper semicontinuous functions with t-norm. Furthermore, the authors also proved several basic properties of convex combination operation and used those to get the strong law of large numbers for pairwise i.i.d. random elements [12,Theorem 5.1], which extended [4, Theorem 1] of Etemadi. Since then, some limit theorems for random elements taking values in convex combination space were considered and extended (see [9,11,12,14]). On the other hand, as shown recently in [13], it is fairly remarkable that although these spaces are not linear in general, they always contains a subspace which can be isometrically embedded into a Banach space and this embedding preserves the convex combination operation.
In this study, we establish the complete convergence for maximum partial sums of a sequence of random elements in a convex combination space, which gives us some new variants of Baum-Katz's type theorems. Notice that some usual techniques developed in Banach space are no longer applicable here because we are dealing with problems in a nonlinear space. For example, Lemma 2.2 in Section 2 is not necessary if one considers the problems in Banach space. Moreover, an illustrative example will be given to show that some conditions appearing in our results cannot be removed in general convex combination space while they become trivial in Banach space. This paper is organized as follows: In Section 2, we state and summarize some basic results about convex combination spaces, discuss the notion of compactly uniform integrability in Cesàro sense and present some auxiliary lemmas. Our main results regarding to Baum-Katz's type theorems for pairwise independent random elements taking values in convex combination space are established in Section 3.

Preliminaries
Throughout this paper, ( , A, P ) is a complete probability space. For A ∈ A, the notation I (A) (or I A ) is the indicator function of A, the symbol C denotes a general positive constant and it is probably not the same in each appearance.
For the reader's convenience, we now present a short introduction to the work given by Terán and Molchanov [12]. Let (X , d) be a metric space. Denote x u := d(u, x) for u, x ∈ X . Based on X , a convex combination operation is defined so that for all n 1, numbers λ 1 , . . . , λ n > 0 that satisfy n i=1 λ i = 1, and all u 1 , . . . , u n ∈ X , this operation produces an element in X , which is denoted by [λ 1 , u 1 ; . . . ; λ n , u n ] or [λ i , u i ] n i=1 . Assume that [1, u] = u for every u ∈ X and the following properties are satisfied: for every permutation σ of {1, . . . , n}; . . . ;λ n , u n ;λ n+1 +λ n+2 , , which is denoted by K X u (or Ku without any confusion), and K is called the convexification operator.
Then, a metric space endowed with a convex combination operation is referred to the convex combination space (shortly, CC space). Notice that [λ 1 , u 1 ; . . . ; λ n , u n ] and the shorthand [λ i , u i ] n i=1 have the same intuitive meaning as the more familiar λ 1 u 1 +· · ·+λ n u n and n i=1 λ i u i , but X is not assumed to have any addition. By induction and (CC.ii), the axiom (CC.iv) can be extended to convex combinations of n elements as follows: The following properties (2.1)-(2.6) are implied from (CC.i)-(CC.v) above, and their proofs were given in [12]: (2.1) For every u 11 , . . . , u mn ∈ X and α 1 , . . . , α m , 2) The convex combination operation is jointly continuous in its 2n arguments. (2.6) The mapping K is non-expansive, that is d(Ku, Kv) d (u, v).
Let λ k ⊂ (0; 1), λ k → 0 and u, v ∈ X . By (CC.iv) and property (2.4), we have When an X -valued random element X takes finite values, it is called a simple random element.
The distribution P X of an X -valued random element X is defined by P X (B) = P (X −1 (B)), ∀B ∈ B(X ), and two X -valued random elements X, Y are said to be identically distributed if P X = P Y . The collection of X -valued random elements {X i , i ∈ I } is said to be independent (resp. pairwise independent) if the collection of σ -algebras In the sequel, we assume that (X , d) is a separable and complete CC space. According to (CC.v), the set K(X ) is nonempty, and hence an element u 0 ∈ K(X ) is fixed. Since X is separable, there exists a countable dense subset {u n , n 1} of X . For each k 1, we define the mapping ϕ k : The expectation for an integrable X -valued random element is constructed via approximation as follows. For a simple random element is an integrable real-valued random variable for some u ∈ X , and the space of all integrable X -valued random elements is denoted by L 1 X . Since X is separable and complete, any integrable X -valued random element can be approximated by a sequence of simple random elements. Namely, if X ∈ L 1 X then X = lim k→∞ ϕ k (X), and the expectation of X is defined by EX := lim k→∞ Eϕ k (X). Based on the approximation, we can also ∈ A for all u i ∈ A and any positive numbers λ i that sum up to 1. The convex hull of A ⊂ X , denoted by coA, is the smallest convex subset of X containing A, and coA denotes the closed convex hull of A. Let k(X ) denote the set of nonempty compact subsets of X . It follows from [12, Theorem 6.2] that if X is a separable complete CC space, then the space k(X ) with the convex combination is a separable complete CC space as well, where the convexification operator K k(X ) is given by This is a nice feature of CC space. Based on this property, if a result holds in CC space, then it can be uplifted to the space of nonempty compact subsets. Further details can be found in [11][12][13].
The notion of compactly uniform integrability in Cesàro sense for a collection of random elements taking values in Banach spaces was discussed by many authors (see, e.g., [1,3,15]). We now introduce this notion in metric spaces, which is also naturally extended from Banach spaces. Let r > 0, then a sequence {X n , n 1} of X -valued random elements is said to be compactly uniformly r-th order integrable in Cesàro sense (Cesàro r-th CUI) if there is a u ∈ X such that for every ε > 0, there exists a compact subset K ε (depending on When r = 1, we also use the terminology Cesàro compactly uniformly integrable or Cesàro CUI for the sake of simplicity. The following proposition shows that the notion of Cesàro r-th CUI does not depend on the selection of u.
1} of X -valued random elements is Cesàro r-th CUI with respect to some element u if and only if it is Cesàro r-th CUI with respect to any element a ∈ X .
Proof Suppose that {X n , n 1} is Cesàro r-th CUI with respect to u and let a be another element of X . For ε > 0 and for each m ∈ N, there exists K ε,m ∈ k(X ) such that denote the open ball with center x ∈ X and radius δ. We have The compactness of K ε,m follows that it can be covered by a finite number of open balls with equal radii m −1/r , and so is Denote K ε = clK ε ∪ K ε,1 , then K ε ∈ k(X ). Hence for all n 1, By the arbitrariness of ε > 0, the proof is complete.
By Jensen's inequality, it is easy to see that for 0 < r p, and this implies that if {X n , n 1} is Cesàro p-th CUI, then it is also Cesàro r-th CUI for 0 < r p. Further details about CUI, the readers can refer to [15]. , u), where x 1 , . . . , x n , u ∈ X are arbitrary.

Baum-Katz's Type Theorems for Pairwise Independent X -Valued Random Elements
Throughout this section, X is a complete and separable CC space and u 0 is the fixed element of K(X ) as mentioned in Section 2.
In the first theorem, we establish a similar result to [1, Theorem 1.2] in CC space. However, the version in CC space has a significant difference compared to the corresponding version in Banach space that is condition (3.1) below. It becomes trivial when one considers in Banach space with usual convex combination operation; moreover, we also show immediately after the proof that it cannot be removed in general CC space, even when considered random elements are independent. We now prove that (3.2) holds.
Step 1 Assume that X is simple with values x 1 , x 2 , . . . , x m on non-null sets 1 , 2 , . . . , m respectively. Since {X, X n , n 1} is identically distributed, each X n also takes values where M := max 1 j m x j u 0 . It implies Next, we show that (I 1 ) < ε/2 for all ω ∈ when n is sufficiently large. Indeed, by the definition of the operator K, for each j = 1, . . . , m. Thus, there exists n 0 (ε, m) ∈ N such that for all n n 0 (ε, m) and for all j = 1, . . . , m, Hence, for n n(ε, m) This implies that for all n n(ε, m). Therefore, ∞ n=1 n p−2 P (( .
Combining above arguments, which means that (3.2) holds for this case.
Step 2 Let us consider the general case when X ∈ L 1 X . For ε > 0 arbitrarily, [12,Proposition 4.1] implies that there exists a natural number h large enough such that Ed(ϕ h (X), X) ε/6, where the function ϕ h was mentioned in Section 2. Then {ϕ h (X), ϕ h (X n ), n 1} is a collection of pairwise i.i.d. and simple random elements with common expectation Eϕ h (X) := b. Moreover, It follows from the first case that ∞ n=1 n p−2 P max Similarly, {d(X, ϕ h (X)), d(X n , ϕ h (X n )), n 1} is also a collection of pairwise i.i.d. realvalued r.v.'s satisfying As a corollary of [1, Theorem By the triangle inequality, and this completes the necessity part.
Sufficiency: Suppose that Ed p (X, KX) < ∞ and (3.2) holds for some a ∈ K(X ). By the triangle inequality, This is equivalent to KX n a d(X n , KX n ) λ i x i , then the condition (3.1) is trivial due to KX = X. However, in general CC space, the condition (3.1) cannot be removed as shown in the example below: Example 3.3 Let p, r be real numbers such that 1 < r < p < 2, r(r − 1) = 1 and r 2 > p. Assume that (X , . ) is a Rademacher type r Banach space and denote by d the metric associated with its norm . . An operation r [., .] is defined based on X as follows: As shown in [12,Example 5], r [., .] is the convex combination operation (r-th power combination) and the corresponding convexification operator K r u = 0 for all u ∈ X . Assume that {X, X n , n 1} is a collection of i.i.d. X -valued random elements satisfying E X r < ∞ but E X p = ∞. Now we show that the condition (3.2) holds while (3.1) does not. It follows from K r X = 0 that Ed p (X, K r X) = E X p = ∞, thus (3.1) fails. Since E X 1 + E X r < ∞, there exists the expectation of X with respect to r [., .], denoted by E r X, and a := E r X = 0. Now for ε > 0, applying the Hájek-Rényi inequality for the collection {X, X n , n 1} of i.i.d. X -valued random elements with EX = 0 as in [10], which means that (3.2) holds.
Next, we establish some results on complete convergence and L r -convergence for nonidentically distributed random elements. Proposition 3.6 Let {X n , n 1} be a sequence of pairwise independent X -valued random elements. If there is a compact subset K of X such that P (X n ∈ K) = 1 for all n, then the following statements hold: Proof For ε > 0, by the compactness of K, there exists {c 1 , c 2 , . . . , c m } ⊂ K such that For n 1, define a sequence of X -valued random elements as follows: It is obvious that the sequence {Y n , n 1} is also pairwise independent. By the triangle inequality, Modifying the proof of [14, Proposition 3.1], we obtain ess sup ω∈ A n (ω) ε/4, ess sup ω∈ D n (ω) ε/4 for all n. Using the same arguments as in the proof of (I 1 ) in Theorem 3.1 (necessity part) to get ess sup ω∈ B n (ω) ε/4 for n large enough. For C n , by property (2.1) and Lemma 2.2, we have where M := max 1 j m c j u 0 . Letting n → ∞ and by the arbitrariness of ε > 0, we derive the desired conclusion. Proof Given ε > 0. By Cesàro r-th CUI assumption (with r = 1 in the case (a)), there exists a compact subset K := K ε,u of X satisfying n −1 n i=1 E( X i r u I (X i / ∈ K)) ε/4 for all n.
For each n 1, we define a sequence of X -valued random elements by setting Y n (ω) = X n (ω) if X n (ω) ∈ K u if X n (ω) / ∈ K.