Law of Large Numbers and Central Limit Theorem Under Probability Uncertainty

Abstract

In this chapter, we first introduce two types of fundamentally important distributions, namely, maximal distribution and a new type of nonlinear normal distribution—G-normal distribution in the theory of sublinear expectations. The former corresponds to constants and the latter corresponds to normal distribution in the classical probability theory. We then present the law of large numbers (LLN) and central limit theorem (CLT) under sublinear expectations. It is worth pointing out that the limit in LLN is a maximal distribution and the limit in CLT is a G-normal distribution.

Notes and Comments

The material of this chapter is mainly from [139, 140, 142] (see also in the Notes [144] in which a stronger condition of the form

$$ \mathbb {E}[|X_1|^p]<\infty ,\,\,\,\, \mathbb {E}[|Y_1|^q]<\infty ,\,\,\,\, \text { for a large }p>2 \text { and }q>1 $$

was in the place of the actual more general Condition (2.4.6). Condition (2.4.6) was proposed in Zhang [180] . The actual proof is a minor technique modification of the original one. We also mention that Chen considered strong laws of large numbers for sublinear expectations in [27]  .

The notion of G-normal distribution was firstly introduced for in [138] for 1-dimen-sional case, and then in [141] for multi-dimensional case. In the classical situation, it is known that a distribution satisfying relation (2.2.1) is stable (see Lévy [107, 108] ). In this sense, G-normal distribution, together with maximal-distribution, are most typical stable distributions in the framework of sublinear expectations.

Marinacci [115]  proposed different notions of distributions and independence via capacity and the corresponding Choquet expectation to obtain a law of large numbers and a central limit theorem for non-additive probabilities (see also Maccheroni  and Marinacci [116]). In fact, our results show that the limit in CLT, under uncertainty, is a G-normal distribution in which the distribution uncertainty cannot be just a family of classical normal distributions with different parameters (see Exercise 2.5.5).

The notion of viscosity solutions plays a basic role in the definitions and properties of G-normal distribution and maximal distribution. This notion was initially introduced by Crandall and Lions  [38]. This is a fundamentally important notion in the theory of nonlinear parabolic and elliptic PDEs. Readers are referred to Crandall et al. [39] for rich references of the beautiful and powerful theory of viscosity solutions. Regarding books on the theory of viscosity solutions and the related HJB equations, see Barles [9], Fleming and Soner  [66] as well as Yong and Zhou [177].

We note that, in the case when the uniform ellipticity condition holds, the viscosity solution (2.2.10) becomes a classical $C^{1+\frac{\alpha }{2} , 2+\alpha }$-solution (see the very profound result of Krylov [105] and in Cabre and Caffarelli  [24] and Wang [171] ). In 1-dimensional situation, when $\underline{\sigma }^{2}>0$, the G-equation becomes the following Barenblatt equation:

$$\begin{aligned} \partial _tu+\gamma |\partial _t u|=\triangle u,~ |\gamma |<1. \end{aligned}$$

This equation was first introduced by Barenblatt [8] (see also Avellaneda et al. [6]).

The rate of convergence of LLN and CLT under sublinear expectation plays a crucially important role in the statistical analysis for random data under uncertainty. We refer to Fang et al. [63], and Song [166, 167], in which a nonlinear generalization of Stein method, obtained by Hu et al. [83], is applied as a sharp tool to attack this problem. A very recent important contribution to the convergence rate of G-CLT is Krylov [106].

We also refer to Jin and Peng [99] for a design of unbiased optimal estimators, as well as [149] for the construction of G-VaR.