Generalized definition of the geometric mean of a non negative random variable

Abstract

The first probabilistic definition of the geometric mean of a non negative random variable under certain assumptions was given in Feng et al. (2013 Feng, C., Wang, H., Tu, X. (2013). Geometric mean of nonnegative random variable. Commun. Stat.—Theory Methods 42:2714–2717.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). In this paper, we generalize the definition to a larger class of random variables. We also show the basic properties of the geometric mean and point out its discontinuity and instability. Some convergence properties are studied as well, for which we emphasize its link to the positive moments of the random variable. A discussion of potential applications of the new definition in biomedical research and open questions to complete the theory of geometric mean is highlighted.