Precise Asymptotics in the Law of the Iterated Logarithm under Sublinear Expectations

By an inequality of partial sum and uniform convergence of the central limit theorem under sublinear expectations, we establish precise asymptotics in the law of the iterated logarithm for independent and identically distributed random variables under sublinear expectations.


Introduction
Motivated by the work of g-expectation of Peng [1], Peng [2,3] initiated the concept of the sublinear expectation space, which is a powerful tool to model the uncertainty of probability and distribution. We could consider sublinear expectation as an extension of the classical linear expectation. Peng [2,3] constructed the basic framework, investigated basic properties, and proved the law of large number and central limit theorem under sublinear expectations. Motivated by the seminal work of Peng [2,3], more and more limit theorems under sublinear expectation space have been established, which generalize the corresponding fundamental, important limit theorems in probability and statistics. Zhang [4][5][6] proved the exponential inequalities and Rosenthal's inequalities and obtained an extension of the central limit theorem and Donsker's invariance principle under sublinear expectations. Wu [7] established precise asymptotics for complete integral convergence under sublinear expectations. Yu and Wu [8] studied Marcinkiewicz-type complete convergence for weighted sums under sublinear expectations. Wu and Jiang [9] obtained a strong law of large numbers and Chover's law of the iterated logarithm under sublinear expectations. Ma and Wu [10] studied the limiting behavior of weighted sums of extended negatively dependent random variables under sublinear expectations. Xu and Zhang [11,12] studied three series theorem for independent random variables and the law of logarithm for arrays of random variables under sublinear expectations. Chen [13] proved strong laws of large numbers for sublinear expectations. For more results about limit theorems under sublinear expectations, the interested reader could refer to the studies of Hu et al. [14], Fang et al. [15], Kuczmaszewska [16], Wang and Wu [17], Hu and Yang [18], Zhang [19], and references therein.
Precise asymptotics in the law of the iterated logarithm is one of the fundamental problems in probability theory. Many related results have been derived in the probabilistic space. eir results can be found in the work of Gut and Spȃtaru [20]; Zhang [21]; Xiao et al. [22]; Huang et al. [23]; Jiang and Yang [24]; Wu and Wen [25]; Xu et al. [26]; Xu [27,28]; and Xu [29]. However, in sublinear expectations, due to the uncertainty of sublinear expectation and related capacity, the precise asymptotics in the law of the iterated logarithm under sublinear expectations have not been reported. Motivated by the work of Wu [7], Xiao et al. [22], Xu et al. [26], and Xu [29], we try to investigate precise asymptotics in the law of the iterated logarithm under sublinear expectations. e aim of this paper is to prove the precise asymptotics in the law of the iterated logarithm for independent, identically distributed random variables under sublinear expectations. e main contribution of this paper is that we prove an useful inequality under sublinear expectations in Lemma 1, and we extend the results of Xiao et al. [22], Xu et al. [26], and Xu [29] to those of the sublinear expectation spaces. Our results may have the potential applications in finance or engineering fields (cf. Wu [7], Peng [3], Zhang [19], and references therein). Our basic idea in this paper comes from that of Wu [7], Xiao et al. [22], Xu et al. [26], Xu [29], Spȃtaru [30], and Fuk and Nagaev [31]. In conclusion, our results combined with the work of Wu [7] imply heuristically that many results about precise asymptotics in the law of the iterated logarithm in probability spaces may still hold under sublinear expectations. e rest of this paper is organized as follows: in Section 2, we summarize necessary basic notions, concepts, and relevant properties and give necessary lemmas under sublinear expectations. In Section 3, we give our main results, eorems 1 and 2, whose proofs are presented in Sections 4 and 5, respectively.

Preliminaries
We use notations similar to those of Peng [3]. Let (Ω, F) be a given measurable space. Let H be a subset of all random variables on (Ω, F) such that for some C > 0, m ∈ N, depending on φ. We regard H as the space of random variables.

Definition 1.
A sublinear expectation E on H is a functional E: H↦R: � [−∞, ∞] satisfying the following properties: for all X, Y ∈ H, we have the following: is called a capacity if it satisfies the following: In this paper, given a sublinear expectation space (Ω, H, E), we define a capacity: Zhang [4]). Clearly, V is a subadditive capacity. We also define the Choquet expectations C V by A sublinear expectation E: H↦R is said to be continuous if it satisfies the following: is said to be continuous capacity if it satisfies the following: Suppose that X 1 and X 2 are two n-dimensional random vectors defined, respectively, in sublinear expectation spaces whenever the sublinear expectations are finite. X n ∞ n�1 is said to be identically distributed if for each i ≥ 1, X i and X 1 are identically distributed.
For 0 ≤ σ 2 ≤ σ 2 < ∞, a random variable ξ under a sub- is the unique viscosity solution of the following heat equation: In the rest of this paper, let {X, X n , n ≥ 1} be a sequence of i.i.d. random variables under sublinear expectation space and −E(−ξ 2 ) � σ 2 . We denote by C a positive constant which may vary from line to line.
To prove our results, we need the following lemmas.
Proof. We borrow the proofs from those of eorem 2 by Fuk and Nagaev [31], and Lemma 2 by Spȃtaru [30]. Let erefore, by the subadditivity property of V(·), By Markov's inequality under sublinear expectations, for any positive h, From this and (7), it follows that Application of the monotonicity of u − 2 (e hu − 1 − hu) for u ≤ y and u − α (e hu − 1 − hu) for u > 0 and the subadditivity property of sublinear expectations yields Hence, by Lemma 1.1 in the study of Gao and Xu [32], Setting in the right-hand side of (11), we see that Since E(X) � E(−X) � 0, by Proposition 3.6 in the study of Peng [3] and Definition 1, we see that erefore, Combining this with (13) and (9), we conclude that Combining (16) with the inequality derived from it with −X and −X k in place of X and X k , respectively, leads to (5). □ Remark 1. (see Lemma 2 in [7]). For any X ∈ H, we have Lemma 2 (see Lemma 5 in [7]). Assume that X n ; n ≥ 1 is a sequence of independent and identically distributed random

Main Results
e following are our main results.

Proof of Theorem 1
Proof.
us, this completes the proof of Proposition 1.
□ Remark 2. By the proof of (24) and (25) in the study by Wu [7], Proof. By Lemma 2 and Toeplitz's lemma, e proof is complete.
Proof. We could obtain that Note that as M ⟶ ∞. Proposition 3 is established.
Mathematical Problems in Engineering Proof. When 0 < b < 2 d, by Markov's inequality under sublinear expectations, we have For b ≥ 2 d, by Lemma 1, we see that where T is a positive constant to be specified later. On the one hand, we obtain that for any T > b/ (2 d). On the other hand, for L 1 , without loss of generality, set T � 1. By the countable subadditivity property of sublinear expectations and the fact that Hence, for b ≥ 2 d, we have us, (28) holds for each b, d > 0. Now, by Proposition 1-4 and the triangle inequality, ∀β > 0, ∃M > 0, which is sufficiently large, such that Mathematical Problems in Engineering 5 We derive eorem 1 from the arbitrariness of β > 0. □

Proof of Theorem 2
Proposition 5. For d > 0, we have Proof. We claim that Indeed, by Lemma 4 in the study by Wu [7], ∀α > 0, Proof. By Lemma 2 and Toeplitz's lemma, 1 y log y log logy V |ξ| ≥ ε(log logy) d dy e proof is complete. Finally, similar to the proof of eorem 1, by the triangle inequality and Propositions 5-8, we finish the proof of eorem 2.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.