Strong law of large numbers for $\varphi$-sub-Gaussian random variables under sub-linear expectation spaces

Abstract

We introduce the notions of sub Gaussian random variables in sub-linear expectation spaces. To avoid the problem caused by the existence of two different expectations, i.e., the upper expectation and the lower expectation, we divide the definition of the sub-Gaussian property into an upper part and a lower part. It turns out that this approach fits well to the sub-linear setting; it provides a proper framework for extending Zajkowski's general result to sublinear expectation spaces. Within our framework, we establish a strong law of large numbers for sub-Gaussian sequences. We present an example showing the usefulness of our results.