P. Terán (2014). Transactions of the American Mathematical Society 366, 5431-5451.
In this paper, a law of large numbers is presented in which the probability measure is replaced by a set function satisfying weaker properties. Instead of the union-intersection formula of probabilities, only complete monotony (a one-sided variant) is assumed. Further, the continuity property for increasing sequences is assumed to hold for open sets but not for general Borel sets.
Many results of this kind have been published in the last years, with very heterogenous assumptions. This seems to be the first result where no extra assumption is placed on the random variables (beyond integrability, of course).
The paper also presents a number of examples showing that the behaviour of random variables in non-additive probability spaces can be quite different. For example, the sample averages of a sequence of i.i.d. variables with values in [0,2] can
-converge almost surely to 0
-have `probability' 0 of being smaller than 1
-converge in law to a non-additive distribution supported by the whole interval [0,1].
Up the line:
This starts a new line.
Down the line:
·Non-additive probabilities and the laws of large numbers (plenary lecture 2011).
To download, click on the title or here.