The law of large numbers (LLN) and central limit theorem (CLT) are long and
widely been known as two fundamental results in probability theory.
<br />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
“% $G$-expectation”, and the related
“$G$-normal distributionfrom 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.
%0 Generic
%1 citeulike:1106690
%A Peng, Shige
%D 2007
%K and central expectations large law limit nonlinear numbers theorem
%T Law of Large Numbers and Central Limit Theorem under Nonlinear Expectations
%U http://arxiv.org/abs/math.PR/0702358
%X The law of large numbers (LLN) and central limit theorem (CLT) are long and
widely been known as two fundamental results in probability theory.
<br />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
“% $G$-expectation”, and the related
“$G$-normal distributionfrom 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.
@misc{citeulike:1106690,
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.
<br />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.},
added-at = {2007-08-18T13:22:24.000+0200},
author = {Peng, Shige},
biburl = {https://www.bibsonomy.org/bibtex/2b05db2d12cfbb7364e02e65604fd1759/a_olympia},
citeulike-article-id = {1106690},
description = {citeulike},
eprint = {math.PR/0702358},
interhash = {5e1716c0e2b223174249f294c30b9ed3},
intrahash = {b05db2d12cfbb7364e02e65604fd1759},
keywords = {and central expectations large law limit nonlinear numbers theorem},
month = Feb,
priority = {2},
timestamp = {2007-08-18T13:22:31.000+0200},
title = {Law of Large Numbers and Central Limit Theorem under Nonlinear Expectations},
url = {http://arxiv.org/abs/math.PR/0702358},
year = 2007
}