The Law of Logarithm for Arrays of Random Variables under Sub-linear Expectations
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Acta Mathemacae Applicatae Sinica, English Series The Editorial Office of AMAS & Springer-Verlag GmbH Germany 2020
The Law of Logarithm for Arrays of Random Variables under Sub-linear Expectations Jia-pan XU1 , Li-xin ZHANG1,† 1 School
of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China (E-mail: [email protected])
Abstract Under the framework of sub-linear expectation initiated by Peng, motivated by the concept of extended negative dependence, we establish a law of logarithm for arrays of row-wise extended negatively dependent random variables under weak conditions. Besides, the law of logarithm for independent and identically distributed arrays is derived more precisely and the sufficient and necessary conditions for the law of logarithm are obtained.
Keywords of logarithm
sub-linear expectation; capacity; exponential inequality; extended negative dependence; the law
2000 MR Subject Classification
1
60F15
Introduction
The emergence of non-additive probabilities and non-additive expectations provides useful tools for studying uncertainties in statistics, measures of risk, superhedging in finance and non-linear stochastic calculus, see [4, 5, 9–13], etc. This paper considers the general sub-linear expectations and related non-additive probabilities generated by them. The general framework of the sublinear expectation was introduced by [12, 14, 15] in a general function space by relaxing the linear property of the classical expectation to the sub-additivity and positive homogeneity. The sublinear expectation does not depend on the probability measure and produces many interesting properties different from those of the linear expectations. Under Peng’s framework, many limit theorems have been being gradually established recently, including the central limit theorem and weak law of large numbers (cf. [14, 16]), the small derivation and Chung’s law of the iterated logarithm (cf. [18]), the strong law of large numbers (cf. [1, 20]). As for the law of the iterated logarithm, Chen[2] gives a result for bounded independent and identically distributed (i.i.d. for short) random variables. Zhang[19] improves the result and extends to independent or negatively dependent identically distributed random variables with only finite second order moments. Moreover, Zhang[21] introduces a concept of extended negative dependence under the sub-linear expectation and shows the corresponding law of iterated logarithm under the condition of finite (2 + α)-moments. Those theorems mentioned above are all about random sequences. However, many important statistical problems are concerned with arrays of random variables, thus it is natural to consider the strong limits for arrays. In classical probability theory, for an array of i.i.d. random variables {Xnk , k = 1, 2, · · · , n; n ∈ N}, Hu et al.[6] obtained the strong law of large numbers Manuscript received December 22, 2018. Accepted on January 17, 2020. This paper is supported by grants from the National Natural Science Foundation of China (No. 11731012), Ten Thousan
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