The Convergence of the Sums of Independent Random Variables Under the Sub-linear Expectations

• Published in: Acta mathematica Sinica. English series Vol. 36; no. 3; pp. 224 - 244
• Main Author: Zhang, Li Xin
• Format: Journal Article
• Language: English
• Published: Beijing Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 01.03.2020; Springer Nature B.V
• Subjects: Central limit theorem; Convergence; Independent variables; Inequalities; Mathematical analysis; Mathematics; Mathematics and Statistics; Normal distribution; Random variables; Sub-linear expectation; central limit theorem; 60G50; independence; Lévy maximal inequality; 60F05; capacity; 60F15
• ISSN: 1439-8516; 1439-7617
• DOI: 10.1007/s10114-020-8508-0

Let { X n ; n ≥ 1} be a sequence of independent random variables on a probability space (Ω ,F ,P) and . It is well-known that the almost sure convergence, the convergence in probability and the convergence in distribution of S n are equivalent. In this paper, we prove similar results for the independent random variables under the sub-linear expectations, and give a group of sufficient and necessary conditions for these convergence. For proving the results, the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established. As an application of the maximal inequalities, the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained.