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

By using Lebesgue bounded convergence theorem, we prove precise asymptotics in the law of the iterated logarithm for independent and identically distributed random variables under sublinear expectation.


Introduction
Inspired by the phenomena of uncertainty in risk measure, super-hedging problems, Peng et al. [1,2] initiated the sublinear expectations space nontrivially. Motivated by the works of Peng, people try to investigate whether or not the corresponding limit theorems in classic probability space could hold under sublinear expectations space. For the corresponding works under sublinear expectations, the interested readers could refer to Gao and Xu [3], Zhang et al. [4][5][6][7][8][9][10], Wu [11], Xu and Cheng [12], Zhong and Wu [13], Xu et al. [14,15], Yu and Wu [16], Wu and Jiang [17], Ma and Wu [18], Chen [19], Fang et al. [20], Hu et al. [21], Hu and Yang [22], Kuczmaszewska [23], Wang and Wu [24], and references therein. Since Gut and Spȃtaru [25] investigated precise asymptotics in the law of the iterated logarithm, people obtained lots of corresponding results related to precise asymptotics in the law of the iterated logarithm, for which the interested readers could refer to Zhang [26], Huang et al. [27], Jiang and Yang [28], Wu and Wen [29], Xiao et al. [30], and Xu et al. [31][32][33]. Recently, by using Lebesgue bounded convergence theorem together with the results of Zhang et al. [8,9], Zhang [10] obtained Heyde's theorem under sublinear expectations. e methods of Zhang [10] are di erent from that of Xiao et al. [30]. By using the central limit theorem obtained by Wu [11] and the methods of Xiao et al. [30], we investigated precise asymptotics in the law of the iterated logarithm under sublinear expectations in Xu and Cheng [12], which complement that of Xiao et al. [30] under sublinear expectations. It is interesting to wonder whether or not precise asymptotics in the law of the iterated logarithm under sublinear expectations hold under the same conditions as in Zhang [10]. In this note, we try to use the methods in Zhang [10] to investigate precise asymptotics in the law of the iterated logarithm for independent and identically distributed random variables under sublinear expectations under conditions di erent from that of Xu and Cheng [12], which complement that obtained by Xu and Cheng [12]. Our results may have potential applications in engineering ornance elds (cf. Peng et al. [2,34,35]). e rest of this paper is organized as follows. In the next section, 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, and we present the proof of eorem 1 in Section 4. e proof of eorem 2 is similar to that of eorem 1, so it is omitted.
φ ∈ C l,Lip (R n ), where C l,Lip (R n ) stands for the linear space of (local Lipschitz) function φ, satisfying 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: In this note, given a sublinear expectation space (Ω, H, E), capacity is defined as follows: Zhang [5]). Clearly, V is a subadditive capacity. We also define the Choquet expectations C V by Assume that X 1 and X 2 are two n-dimensional random vectors defined, respectively, in sublinear expectation spaces ey are called to be identically distributed if Whenever the sublinear expectations are finite. X n ∞ n�1 is called to be identically distributed if for each i ≥ 1, X i and X 1 are identically distributed.
Lemma 2 (see Lemma 2.4 in [10]). Assume that X n ; n ≥ 1 is a sequence of independent, identically distributed random variables in a sublinear expectation space where ξ ∼ N(0, [r, 1]), r � σ 2 /σ 2 . Moreover, for any continuous and bounded functions ψ: where and where and N is a standard random variable.
In the remainder of this paper, let\{X, X n , n ≥ 1\} be a sequence of i.i.d. random variables under sublinear expec- Let log x � ln(x∨e), log logx � ln(ln(x∨e e )), and ⌊ x ⌋ � sup l, l ≤ x, l ∈ Z + { }; we denote by C a positive constant which may change from line to line.

Main Results
e following are our main results.

Proof of Theorem 1
Here, we borrow ideas from the proofs in Zhang [10]. Without loss of restrictions, we suppose σ � 1. We first establish (11). Notice for n large enough, Mathematical Problems in Engineering 3 erefore, it is enough to establish that Obviously, we see that for any δ > 0, For M > 1 ≥ δ > ε > 0, and 0 ≤ a ≤ 1, As in the proofs of Zhang [10], C V (X 2 1 ) < ∞ implies the assumptions of Lemma 2. By Lemma 2 and Corollary 1, If z d/b is the continuous point of F. Note that F(x) is a monotone function, whose discontinuous points are countable, z d/b is one-to-one mapping from [0, ∞) to [0, ∞), and so the above convergence holds except on a set with null Lebesgue measure. From the Lebesgue bounded convergence theorem it follows that lim ε↘0 Mathematical Problems in Engineering On the other hand, for M ≥ 2, by the study (24) and (25) in [11], It remains to establish that Hence, we see that for (M 2 d /2) > 32 2 and x ≥ b(ε), erefore, by the proof of Lemma 1 in Zhong and Wu [13], Mathematical Problems in Engineering As M ⟶ ∞, for (−S k ) s, we have same convergence. us, (11) is established.
Next, we prove (12). With (9) in place of (8), we obtain Also, for 0 ≤ a ≤ 1, We see that for any integer m ≥ 0, Hence, letting m ⟶ ∞ in the above two inequalities results in erefore, (12) is proved.

Data Availability
No data were used to support this study.

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