ABSTRACT
Recently, the law of large numbers (SLLN for short) and the central limit theorem (CLT for short) were given under sublinear expectations by Peng in 2007 (see Peng 2007a,b, 2008-2010). Later, Chen (2010) investigated a strong law of large numbers (SLLN for short) under capacities generated by a sublinear expectation. He showed that for a sequence of independent and identically distributed (IID for short) random variables {Xn}∞n=1 under a sublinear expectation E˜, if E˜[|X1|1+α] <∞ for someα∈ (0, 1), and there exist two real constants µ1 ≤µ2 such that −E˜[−X1] =µ1 and E˜[X1] =µ2, then we have
where v(A) =−E˜[−IA], ∀A∈F , is a lower probability on a measurable space (,F), Sn = ∑ni=1 Xi, and if we further assume the upper probabilityV (A) = E˜[IA], ∀A∈F , is continuous, then for any x ∈ [µ1, µ2], we haveV (x ∈C( Snn )) = 1, whereC(xn) is the cluster set of limit points of a real sequence {xn}∞n=1. For more results of strong law of large numbers under capacities one can refer to Marinacci (1999), Maccheroni & Marinacci (2005), Chen & Wu (2011), Hu (2012), Chen (2012) and the references therein.
