Marcinkiewicz–Zygmund Laws of Large Numbers under Sublinear Expectation

+e theory of sublinear expectation was initiated by Peng [1, 2] to describe the probability uncertainties in statistics, economics, finance, and other fields which are difficult to be handled by the classical probability theory.+e classical laws of large numbers (LLNs for short) which reveal the almost sure laws of stabilized partial sum are of great significance in the probability theory. Recently, the LLNs under sublinear expectation got a lot of development, see for example, Marinacci [3]; Maccheroni and Marinacci [4]; Chen et al. [5]; Chen [6]; Zhang [7]; Hu [8]; Chen et al. [9]; and Hu [10]. Peng [11]; Chen et al. [9]; and Hu [12] gave three forms of weak LLNs under sublinear expectation under the first moment condition. +at is, for any φ ∈ Cb(R),


Introduction
e theory of sublinear expectation was initiated by Peng [1,2] to describe the probability uncertainties in statistics, economics, finance, and other fields which are difficult to be handled by the classical probability theory. e classical laws of large numbers (LLNs for short) which reveal the almost sure laws of stabilized partial sum are of great significance in the probability theory. Recently, the LLNs under sublinear expectation got a lot of development, see for example, Marinacci [3]; Maccheroni and Marinacci [4]; Chen et al. [5]; Chen [6]; Zhang [7]; Hu [8]; Chen et al. [9]; and Hu [10].
Peng [11]; Chen et al. [9]; and Hu [12] gave three forms of weak LLNs under sublinear expectation under the first moment condition. at is, for any φ ∈ C b (R), for any ε > 0, and for any h ∈ [μ, μ] where (E, E) denotes the pair of the sublinear expectation and the conjugate expectation and (V, v) represent the induced capacities, μ � E[X 1 ] and μ � E[X 1 ].
Chen et al. [5] obtained the Kolmogorov strong LLNs under sublinear expectation under the condition of finite (1 + α)th moments (α > 0) for sublinear expectation: Zhang [7] got the above strong LLNs under the condition of finite first moment for Choquet expectation. Hu [8] improved the above results under a general moment condition for sublinear expectation which is the weakest one for sublinear expectation. e classical Marcinkiewicz-Zygmund strong LLNs generalized the Kolmogorov strong LLNs by extending the convergence rate of partial sum and give the relation between moment conditions and convergence rate. e norming constants become n 1/p (0 < p ≤ 2) instead of n and the moment conditions depend on p accordingly. In sublinear situation, Feng and Lan [13] obtained the Marcinkiewicz-Zygmund strong LLNs for arrays of row wise independent random variables. Zhang and Lin [14]; Xu and Zhang [15] got the Marcinkiewicz-Zygmund strong LLNs under the condition of finite pth moments for Choquet expectation by different methods. We know that Choquet expectation is larger than sublinear expectation. e purpose of this paper is to generalize the weak and strong LLNs to the Marcinkiewicz-Zygmund LLNs under some moment conditions for sublinear expectation. We discuss the weak results under the condition of porder uniform integrability for sublinear expectation and study the strong results under the condition a bit stronger than finite pth moments for sublinear expectation. e plan of this paper is as follows. In Section 2, we introduce the basic concepts and lemmas under sublinear expectation. In Section 3, we prove some forms of Marcinkiewicz-Zygmund weak LLNs under sublinear expectation and discuss the equivalence relation among them. In Section 4, the Marcinkiewicz-Zygmund strong LLNs under sublinear expectation is given.

Preliminaries
Let (Ω, F) be a measurable space and H be a linear space of real functions defined on (Ω, F) such that if for some C > 0.
Remark 2. Let P be a family of probability measures defined on (Ω, F). For any random variable X ∈ F, the upper expectation defined by E[X]: � sup Q∈P E Q [X] is a sublinear expectation. So, the results in this paper can also be applied to upper expectation. e conjugate expectation of E is defined by Obviously, for all X ∈ H, E[X] ≤ E[X].
Definition 2 (see [16]). A set function V: In this paper, we consider the capacities induced by sublinear expectation and the conjugate expectation: e continuity from above and continuity from below of sublinear expectation and capacity can also be defined similar to the classical probability theory (see [7]).
whenever the sublinear expectations on both sides are finite. X n ∞ n�1 is said to be a sequence of independent random variables, if X n+1 is independent of (X 1 , . . . , X n ) for each n ≥ 1.

Remark 3.
Peng [2] also gave the definition of identical distribution under the sublinear expectation space. e results in this paper do not need the random variables to be identically distributed.
To prove our main results, we need the following lemmas. e proofs of Lemma 1, Lemma 2, and Lemma 3 can be found in [5] and [8].

Lemma 1 (Borel-Cantelli lemma). If sublinear expectation E is continuous from below and
Lemma 2 (Chebyshev's inequality). Let f(x) > 0 be a nondecreasing function on R. en, for any real x, In the following sections, we consider the sequence X n ∞ n�1 of independent random variables defined on a sublinear expectation space
Remark 4. If sublinear expectation E coincides with the classical expectation E P where P is classical probability, (11)-(13) and (34) all can be reduced to the classical forms.
(2) For any fixed a > 0, there exists C > 0 such that ψ(x + a) ≤ Cψ(x) for any x > 0. Before we give the main results, we need the following two lemmas which are used to cope with the truncated parts.
en, for any m > 1, we have Proof.

By Lemma 5 and
By Borel-Cantelli lemma, we have e continuity from below of V can be deduced by the continuity from below of E. So, we have where By Lemma 4, we have By Kronecker lemma, we have On the other hand, by the continuity from below of E, Taking lim sup n⟶∞ on both sides of (54), and by (58) Equivalently, Equivalently, Noting that V(A ∪ B) ≤ V(A) + V(B), we obtain (49).
For any ε > 0, we set m > (1/ε). en, by Chebyshev's inequality and Lemma 5, we have By Borel-Cantelli lemma, we have Hence, by the continuity from below of V, we have By Borel-Cantelli lemma, we have en, for any ω ∈ ∪ ∞ n�1 ∩ ∞ i�n B c i , there exists n(ω) ∈ N * such that for any i > n(ω), |X i (ω)| ≤ (i 1/p /ln(1 + i)). For these ω, which implies Taking lim sup n⟶∞ on both sides of (70) and by (73) e two moment conditions for case 1 ≤ p < 2 and case 0 < p < 1 in eorem 3 are both stronger than sup n≥1 E[|X n | p ] < ∞ but weaker than sup n≥1 E[|X n | p+α ] < ∞(α > 0). Zhang and Lin [14] obtained that the pth moments for Choquet expectation is the necessary and sufficient conditions of the Marcinkiewicz-Zygmund strong LLNs. One can find a counterexample that the pth moments for sublinear expectation is finite but the pth moments for Choquet expectation is not finite (similar to Example 4.1 of Hu [12]. So the pth moments for sublinear expectation cannot maintain the Marcinkiewicz-Zygmund strong LLNs. Remark 6. If sublinear expectation E coincides with the classical expectation E P where P is classical probability, eorem 3 can be reduced to the classical Marcinkiewicz-Zygmund strong LLNs: where a � 0 if 0 < p < 1 and a � E[X 1 ] if 1 ≤ p < 2.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest.