A strong law of large numbers for non-additive probabilities

In this paper, with the notion of independence for random variables under upper expectations, we derive a strong law of large numbers for non-additive probabilities. This result is a natural extension of the classical Kolmogorov's strong law of large numbers to the case where the probability is no longer additive. As an application of our result, we give most frequent interpretation for Bernoulli-type experiments with ambiguity.

[1]  Zengjing Chen,et al.  Strong Laws of Large Numbers for Bernoulli Experiments under Ambiguity , 2011, NL-MUA.

[2]  Massimo Marinacci,et al.  A strong law of large numbers for capacities , 2005 .

[3]  Massimo Marinacci,et al.  Limit Laws for Non-additive Probabilities and Their Frequentist Interpretation , 1999 .

[4]  Gert de Cooman,et al.  Imprecise probability trees: Bridging two theories of imprecise probability , 2008, Artif. Intell..

[5]  S. Peng.  Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyers type , 1999 .

[6]  P. Wakker.  Testing and Characterizing Properties of Nonadditive Measures through Violations of the Sure-Thing Principle , 2001 .

[7]  S. Peng.  Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation , 2006, math/0601699.

[8]  I. Gilboa.  Expected utility with purely subjective non-additive probabilities , 1987 .

[9]  V. Strassen.  An invariance principle for the law of the iterated logarithm , 1964 .

[10]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[11]  T. Fine,et al.  Towards a Frequentist Theory of Upper and Lower Probability , 1982 .

[12]  D. Schmeidler.  Subjective Probability and Expected Utility without Additivity , 1989 .

[13]  Larry G. Epstein,et al.  Ambiguity, risk, and asset returns in continuous time , 2000 .

[14]  Shige Peng,et al.  Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations , 2009 .

[15]  Shige Peng,et al.  Function Spaces and Capacity Related to a Sublinear Expectation: Application to G-Brownian Motion Paths , 2008, 0802.1240.

[16]  L. Wasserman,et al.  Bayes' Theorem for Choquet Capacities , 1990 .

[17]  Gert de Cooman,et al.  Weak and strong laws of large numbers for coherent lower previsions , 2008 .

[18]  Martin Schneider,et al.  IID: independently and indistinguishably distributed , 2003, J. Econ. Theory.