Laws of large numbers without additivity

Abstract

The law of large numbers is studied under a weakening of the axiomatic properties of a probability measure. Averages do not generally converge to a point, but they are asymptotically confined in a limit set for any random variable satisfying a natural ‘finite first moment’ condition. It is also shown that their behaviour can depart strikingly from the intuitions developed in the additive case.

Sublinear Expectations: On Large Sample Behaviours, Monte Carlo Method, and Coherent Upper Previsions

“…Proposition 5.3 in [8] presents an example of behaviours similar to Proposition 2, but the independence in the sense of Peng is not satisfied there (only a rather weaker form of independence holds), and a different definition of convergence in law is used as well. Theorem 1.…”

“…To make things even more interesting, for this sublinear expectation convergence in probability actually implies almost sure convergence; that is an instance of a more general phenomenon (see e.g. [8,Proposition 5.1]).…”

Extension of the strong law of large numbers for capacities

“…Immediately, for a given random variable X on (Ω, F), two kinds of nonlinear expectations can be denoted via capacities (V, v). One is a pair (C V , C v ) of Choquet integrals corresponding to capacity V = V and V = v, where the Choquet integral with respect to capacity V is defined by With a notion of independence relative to capacity, Maccheroni and Marinacci [17], Marinacci [18], Terán [25] and some of the references therein investigate the strong laws of large numbers via Choquet integrals with restrictive assumptions on sample space Ω and capacity V or v. For instance, Ω is a compact or Polish topological space, or capacity v is completely monotone, or at least 2-monotone. They show that the cluster points of empirical averages lie in the interval [C…”

Weak and Strong Limit Theorems for Stochastic Processes under Nonadditive Probability

“…When { } ∞ =1 has mean uncertainty, sample mean / probably cannot converge to a unique constant almost everywhere (shortly a.e., which should be well defined) under a nonadditive probability or a set of probabilities. Marinacci [8], Teran [9], and some of the references therein investigate the SLLN via Choquet integrals related to completely monotone capacity . They suppose that { } ∞ =1 is a sequence of independent and identically distributed random variables under capacity and prove that all the limit points of convergent subsequences of sample mean / belong to an interval…”