Law of large numbers for m-dependent random vectors under sublinear expectations

Abstract

Sublinear expectation relaxes the linear property of classical expectation to subadditivity and positive homogeneity, which can be expressed as E(⋅)=sup_θ∈ΘE_θ(⋅) for a certain set of linear expectations {E_θ:θ∈Θ}. Such a framework can capture the uncertainty and facilitate a robust method of measuring risk loss reasonably. This study established a law of large numbers for m-dependent random vectors within the framework of sublinear expectation. Consequently, the corresponding explicit rate of convergence were derived. The results of this study can be considered as an extension of the Peng’s law of large numbers [22].

Keywords

Law of large numbers / m-dependence / Sublinear expectations / Rate of convergence / Random vectors

Acknowledgements

We thank the Editor, the Associate Editor and two anonymous reviewers for their helpful suggestions that greatly improve this paper. Cheng’s research was funded by the National Nature Science Foundation of China (Grant No. 12001128) and the GuangDong Basic and Applied Basic Research Foundation (Grant No. 2022A1515011899).

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