Law of Large Numbers and Central Limit Theorem under Nonlinear Expectations

Abstract

The law of large numbers (LLN) and central limit theorem (CLT) are long and widely been known as two fundamental results in probability theory. Recently problems of model uncertainties in statistics, measures of risk and superhedging in finance motivated us to introduce, in [4] and [5] (see also [2], [3] and references herein), a new notion of sublinear expectation, called \textquotedblleft% $G$-expectation\textquotedblright, and the related \textquotedblleft$G$-normal distribution\textquotedblright from which we were able to define G-Brownian motion as well as the corresponding stochastic calculus. The notion of G-normal distribution plays the same important rule in the theory of sublinear expectation as that of normal distribution in the classic probability theory. It is then natural and interesting to ask if we have the corresponding LLN and CLT under a sublinear expectation and, in particular, if the corresponding limit distribution of the CLT is a G-normal distribution. This paper gives an affirmative answer. The proof of our CLT is short since we borrow a deep interior estimate of fully nonlinear PDE in [6] which extended a profound result of [1] (see also [7]) to parabolic PDEs. The assumptions of our LLN and CLT can be still improved. But the discovered phenomenon plays the same important rule in the theory of nonlinear expectation as that of the classical LLN and CLT in classic probability theory.

7 pages

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• G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty
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• Tightness, weak compactness of nonlinear expectations and application to CLT
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• Properties of solutions of stochastic differential equations driven by the G-Brownian motion
• Explicit solutions of G-heat equation with a class of initial conditions by G-Brownian motion
• Moment bounds for IID sequences under sublinear expectations
• G-Gaussian Processes under Sublinear Expectations and q-Brownian Motion in Quantum Mechanics
• Some inequalities and limit theorems under sublinear expectations
• Some apriori estimates of G-BSDEs and the G-martingale representation for a special case
• Multi-dimensional central limit theorems and laws of large numbers under sublinear expectations
• Robust valuation and risk measurement under model uncertainty
• Convergence rates in the law of large numbers under sublinear expectations
• Functional central limit theorems for random vectors under sub-linear expectations
• Almost Sure Central Limit Theorem in Sub-linear Expectation Spaces