Qiu, Dehua; Zhao, Qianjun Convergence rate in the law of logarithm for NA random variables. (Chinese. English summary) Zbl 07809947 Chin. J. Appl. Probab. Stat. 39, No. 5, 659-666 (2023). Summary: Let \(\{X, X_{n}, n \geqslant 1\}\) be a sequence of identically distributed NA random variables and set \(S_{n} = \sum_{i = 1}^{n} X_{i}\), \(n \geqslant 1\). Let \(h (\cdot)\) be a positive nondecreasing function on (\(0, \infty\)) such that \(\int_{1}^{\infty} [th (t)]^{-1} \mathrm{d}t = \infty\). Denote \(Lt = \ln \max \{e, t\}\), \(S_{n} = \sum_{i = 1}^{n} X_{i}\), \(\psi (t) = \int_{1}^{t} [sh (s)]^{-1} \mathrm{d}s\), \(t \geqslant 1\). In this paper, we prove that \(\sum_{n=1}^{\infty} [nh(n)]^{-1} \mathsf{P} (\max_{1 \leqslant j \leqslant n} |S_{j}| \geqslant (1 + \varepsilon) \sigma \sqrt{2nL \psi (n)}) < \infty\), \(\forall \varepsilon > 0\) if and if \(\mathsf{E}(X) = 0\) and \(\mathsf{E}(X^{2}) = \sigma^{2} \in (0,\infty)\). The result partially extends and improves the theorems of D. Li and A. Rosalsky [J. Math. Anal. Appl. 330, No. 2, 1488–1493 (2007; Zbl 1123.60018)]. MSC: 60F15 Strong limit theorems Keywords:NA random variables; law of logarithm; convergence rate Citations:Zbl 1123.60018 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] HARTMAN P, WINTNER A. On the law of the iterated logarithm [J]. Amer J Math, 1941, 63(1): 169-176. · Zbl 0024.15802 [2] DAVIS J A. Convergence rates for the law of the iterated logarithm [J]. Ann Math Statist, 1968, 39(5): 1479-1485. · Zbl 0174.49902 [3] LI D L, WANG X C, RAO M B. Some results on convergence rates for probabilities of moderate deviations for sums of random variables [J]. Internat J Math Math Sci, 1992, 15(3): 481-497. · Zbl 0753.60028 [4] GUT A. Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices [J]. Ann Probab, 1980, 8(2): 298-313. · Zbl 0429.60022 [5] LI D L. Convergence rates of law of iterated logarithm for B-valued random variables [J]. Sci China Ser A, 1991, 34(4): 395-404. · Zbl 0735.60031 [6] CHEN P Y, WANG D C. Convergence rates for probabilities of moderate deviations for moving average processes [J]. Acta Math Sin (Engl Ser), 2008, 24(4): 611-622. · Zbl 1159.60015 [7] LI D L, ROSALSKY A. A supplement to the Davis-Gut law [J]. J Math Anal Appl, 2007, 330(2): 1488-1493. · Zbl 1123.60018 [8] LIU X D, QIAN H Y, CAO L Q. The Davis-Gut law for moving average processes [J]. Statist Probab Lett, 2015, 104: 1-6. · Zbl 1326.60034 [9] FRIEDMAN N, KATZ M, KOOPMANS L H. Convergence rates for the central limit theorem [J]. Proc Nat Acad Sci USA, 1966, 56(4): 1062-1065. · Zbl 0147.17002 [10] ALAM K, SAXENA K M L. Positive dependence in multivariate distributions [J]. Comm Statist Theory Methods, 1981, 10(12): 1183-1196. · Zbl 0471.62045 [11] JOAG-DEV K, PROSCHAN F. Negative association of random variables, with applications [J]. Ann Statist, 1983, 11(1): 286-295. · Zbl 0508.62041 [12] BLOCK H W, SAVITS T H, SHAKED M. Some concepts of negative dependence [J]. Ann Probab, 1982, 10(3): 765-772. · Zbl 0501.62037 [13] SHAO Q M. A comparison theorem on moment inequalities between negatively associated and inde-pendent random variables [J]. J Theoret Probab, 2000, 13(2): 343-356. · Zbl 0971.60015 [14] 苏淳, 王岳宝. 同分布 NA 序列的强收敛性 [J]. 应用概率统计, 1998, 14(2): 131-140. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.