Extension of the strong law of large numbers for capacities

Abstract

In this paper, with a new notion of exponential independence for random variables under an upper expectation, we establish a kind of strong laws of large numbers for capacities. Our limit theorems show that the cluster points of empirical averages not only lie in the interval between the upper expectation and the lower expectation with lower probability one, but such an interval also is the unique smallest interval of all intervals in which the limit points of empirical averages lie with lower probability one. Furthermore, we also show that the cluster points of empirical averages could reach the upper expectation and lower expectation with upper probability one.

Ergodicity of invariant capacities

“…Let (V, v) denote the g-probabilities generated by the backward stochastic differential equations (or g-expectations) with generators g(z) = k|z| and g = −k|z| respectively, where k is any fixed number in R + . Then (V, v) is a pair of continuous upper-lower probabilities (see Example 1 in [5]). Thus, our results in Theorems 4.3, 4.4 and 4.5 apply immediately to this case.…”

Strong laws of large numbers for general random variables in sublinear expectation spaces

“…Since then, many authors began to investigate the limit properties, especially the law of large numbers in sublinear expectation spaces. For instance, Chen, Wu and Li [5] gave a strong law of large numbers for non-negative product independent random variables; Chen [2] obtained the Kolmogorov strong law of large numbers for independent and identically distributed random variables; Zhang [21] gave the necessary and sufficient conditions of Kolmogorov strong law of larger numbers holding for independent and identically distributed random variables under a continuous sublinear expectation; Zhang [22] obtained a strong law of large numbers for a sequence of extended independent random variables; Chen, Hu and Zong [3] gave several strong laws of large numbers under some moment conditions with respect to the partial sum and some conditions similar to Petrov's; Chen, Huang and Wu [4] established an extension of the strong law of large numbers under exponential independence. There are also many results on law of large numbers under different non-additive probability framework.…”

“…Inequalities(2) and(4) are Kolmogorov maximal inequalities in the moment and capacity types, respectively. As well, inequalities (3) and(5) are Hájek-Rényi maximal inequalities in the moment and capacity types, respectively.…”

Almost Sure Central Limit Theorem in Sub-linear Expectation Spaces

“…when {Z l } ∞ l=1 are independent identically distributed under E with E[Z 1 ] = µ and −E[−Z 1 ] = µ. Chen et al [10] established an extension strong law of large numbers under exponential independence. Unfortunately, the {Z l } ∞ l=1 in (4) are neither independent nor exponential independent under E, thus we give a strong law of large numbers without independence but with some moment conditions in section 3.…”