A strong law of large numbers for independent random variables under non-additive probabilities

Abstract

Under non‐additive probabilities, cluster points of the empirical average have been proved to quasi-surely fall into the interval constructed by either the lower and upper expectations or the lower and upper Choquet expectations. In this paper, based on the initiated notion of independence, we obtain a different Marcinkiewicz-Zygmund type strong law of large numbers. Then the Kolmogorov type strong law of large numbers can be derived from it directly, stating that the closed interval between the lower and upper expectations is the smallest one that covers cluster points of the empirical average quasi-surely.

Keywords:

• Independent random variables
• non‐additive probabilities
• strong laws of large numbers
• upper expectations
• weak laws of large numbers

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 60F15; 60F05

Additional information

Funding

Yuting Lan’s work was supported by the Fundamental Research Funds for Central Universities of Shanghai University of Finance and Economics (No. 2017110072) and the National Natural Science Foundation of China (No. 11601280).