Strong laws of large numbers for sub-linear expectations

We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov's strong law of large numbers to the case where probability measures are no longer additive. An important feature of these strong laws of large numbers is to provide a frequentist perspective on capacities.Comment: 10 page

Ergodic Theorems for Lower Probabilities

We establish an Ergodic Theorem for lower probabilities, a generalization of standard probabilities widely used in applications. As a by-product, we provide a version for lower probabilities of the Strong Law of Large Numbers

A strong law of large numbers for capacities

We consider a totally monotone capacity on a Polish space and a sequence of bounded p.i.i.d. random variables. We show that, on a full set, any cluster point of empirical averages lies between the lower and the upper Choquet integrals of the random variables, provided either the random variables or the capacity are continuous.Comment: Published at http://dx.doi.org/10.1214/009117904000001062 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org