Precise Rates in the Law of Iterated Logarithm for the Moment Convergence of φ-Mixing Sequences

Abstract

Let {X

Note on Precise Rates in the Law of Iterated Logarithm for the Moment Convergence of I.I.D.: Random Variables under Sublinear Expectations

“…Remark 1. eorem 1 generalizes eorem 1.1 in [33] in some extent. When the sublinear expectation space is linear and {X n , n ≥ 1} is i. i. d., eorem 1 is the same as (1.5) of eorem 1.1 in [33].…”

“…Next, for J 3 , by Lemma 1, we obtain E(M 2 n ) ≤ Cn. en, by Markov's inequality, under sublinear expectations and Lemma 2.4 of [33], we see that…”

“…Since Gut and Spȃtaru [27] studied precise asymptotics in the law of the iterated logarithm, people established lots of results corresponded to precise asymptotics in the law of the iterated logarithm, for which the readers could refer to Zhang [5,28], Huang et al [29], Jiang and Yang [30], Wu and Wen [31], Xiao et al [32], Xiao and Yin [33], Xu et al [34], and Xu and Cheng [35,36]. Recently, Zhang [5] obtained Heyde's theorem under sublinear expectations and Xiao and Yin [33] proved precise rates in the law of iterated logarithm for the moment convergence of φ-mixing sequence in classic probability space.…”

“…Since Gut and Spȃtaru [27] studied precise asymptotics in the law of the iterated logarithm, people established lots of results corresponded to precise asymptotics in the law of the iterated logarithm, for which the readers could refer to Zhang [5,28], Huang et al [29], Jiang and Yang [30], Wu and Wen [31], Xiao et al [32], Xiao and Yin [33], Xu et al [34], and Xu and Cheng [35,36]. Recently, Zhang [5] obtained Heyde's theorem under sublinear expectations and Xiao and Yin [33] proved precise rates in the law of iterated logarithm for the moment convergence of φ-mixing sequence in classic probability space. In this note, motivated by the works of Zhang [5] and Xiao and Yin [33], we try to prove precise rates in the law of iterated logarithm for the moment convergence of maxima of sums of i. i. d. random variables under sublinear expectations, which complement that of Xiao and Yin [33] under sublinear expectations and that of Xu and Cheng [14].…”

“…Recently, Zhang [5] obtained Heyde's theorem under sublinear expectations and Xiao and Yin [33] proved precise rates in the law of iterated logarithm for the moment convergence of φ-mixing sequence in classic probability space. In this note, motivated by the works of Zhang [5] and Xiao and Yin [33], we try to prove precise rates in the law of iterated logarithm for the moment convergence of maxima of sums of i. i. d. random variables under sublinear expectations, which complement that of Xiao and Yin [33] under sublinear expectations and that of Xu and Cheng [14]. Our result perhaps has potential applications in nature and society (cf.…”