Abstract
In this paper, we introduce the notions of uniform integrability in the Cesàro sense, h-integrability with respect to the array of constants {ani}, and h-integrability with exponent r for an array of measurable operators. Then, we establish some mean convergence theorems and weak laws of large numbers for arrays of measurable operators under some conditions related to these notions.
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Funding
The research of the first two authors was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the grant number is 101.03-2017.24. The research of the third author was supported by the Ministry of Science and Technology, R.O.C. (MOST 105-2118-M-007-004-MY2). The research of the fourth author was supported by Department of Science and Technology (HCMC-DOST) and Institute for Computational Science and Technology (ICST) at Ho Chi Minh City, Vietnam, under grant number 14/2017/HD-KHCNTT.
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Quang, N.V., Son, D.T., Hu, TC. et al. Mean Convergence Theorems and Weak Laws of Large Numbers for Arrays of Measurable Operators under Some Conditions of Uniform Integrability. Lobachevskii J Math 40, 1218–1229 (2019). https://doi.org/10.1134/S1995080219080249
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DOI: https://doi.org/10.1134/S1995080219080249