A general law of the iterated logarithm for non-additive probabilities

Abstract

Motivated by some interesting problems in mathematical economics, quantum mechanics and finance, non-additive probabilities have been used to describe the phenomena which are generally non-additive. In this paper, we further study the law of the iterated logarithm (LIL) for non-additive probabilities, based on existing results. Under the framework of sublinear expectation initiated by Peng, we give two convergence results of $V_n≔\frac{{\sum }_{i=1}^n{X}_i}{\sqrt{n}\varphi \left(n\right)}$ under some reasonable assumptions, where ${\left\{X_i\right\}}_{i=1}^{\infty }$ is a sequence of random variables and $\varphi$ is a positive nondecreasing function. From these, a general LIL for non-additive probabilities is proved for negatively dependent and non-identically distributed random variables. It turns out that our result is a natural extension of the Kolmogorov LIL and the Hartman–Wintner LIL. Theorem 1 and Theorem 2 in this paper can be seen an extension of Theorem 1 in Chen and Hu (2014).