Let $S_n= \sum_{i=1}^n X_i$ where $\{X_i\}_{i=1}^\infty$ is a sequence of independent and identically distributed (i.i.d.) of random variables with $E[X_1]=m$. According to the classical law of large number (LLN), the sum $S_n/n$ converges strongly to $m$. Moreover, the well-known central limit theorem (CLT) tells us that, with $m = 0$ and $s^2=E[X_1^2]$, for each bounded and continuous function $j$ we have $\lim_n E[j(S_n/\sqrt{n}))]=E[j(X)]$ with $X \sim N(0, s^2)$.
These two fundamentally important results are widely used in probability, statistics, data analysis as well as in many practical situation such as financial pricing and risk controls. They provide a strong argument to explain why in practice normal distributions are so widely used. But a serious problem is that the i.i.d. condition is very difficult to be satisfied in practice for the most real-time processes for which the classical trials and samplings becomes impossible and the uncertainty of probabilities and/or distributions cannot be neglected.
In this talk we present a systematical generalization of the above LLN and CLT. Instead of fixing a probability measure P, we only assume that there exists a uncertain subset of probability measures $\{P_q:q \in Q\}$. In this case a robust way to calculate the expectation of a financial loss $X$ is its upper expectation: $[\^\,(\mathbf{E})][X]=\sup_{q \in Q} E_q[X]$ where $E_q$ is the expectation under the probability $P_q$. The corresponding distribution uncertainty of $X$ is given by $F_q(x)=P_q(X \leq x)$, $q \in Q$. Our main assumptions are:
- The distributions of $X_i$ are within an abstract subset of distributions $\{F_q(x):q \in Q\}$, called the distribution uncertainty of $X_i$, with $['(m)]=[\^(\mathbf{E})][X_i]=\sup_q\int_{-\infty}^\infty xF_q(dx)$ and $m=-[\^\,(\mathbf{E})][-X_i]=\inf_q \int_{-\infty}^\infty x F_q(dx)$.
- Any realization of $X_1, \ldots, X_n$ does not change the distributional uncertainty of $X_{n+1}$ (a new type of `independence' ).
Our new LLN is: for each linear growth continuous function $j$ we have
$$\lim_{n\to\infty} \^{\mathbf{E}}[j(S_n/n)] = \sup_{m\leq v\leq ['(m)]} j(v)$$
Namely, the distribution uncertainty of $S_n/n$ is, approximately, $\{ d_v:m \leq v \leq ['(m)]\}$.
In particular, if $m=['(m)]=0$, then $S_n/n$ converges strongly to 0. In this case, if we assume furthermore that $['(s)]2=[\^\,(\mathbf{E})][X_i^2]$ and $s_2=-[\^\,(\mathbf{E})][-X_i^2]$, $i=1, 2, \ldots$. Then we have the following generalization of the CLT:
$$\lim_{n\to\infty} [j(Sn/\sqrt{n})]= \^{\mathbf{E}}[j(X)], L(X)\in N(0,[s^2,\overline{s}^2]).$$
Here $N(0, [s^2, ['(s)]^2])$ stands for a distribution uncertainty subset and $[\^(E)][j(X)]$ its the corresponding upper expectation. The number $[\^(E)][j(X)]$ can be calculated by defining $u(t, x):=[^(\mathbf{E})][j(x+\sqrt{tX})]$ which solves the following PDE $\partial_t u= G(u_{xx})$, with $G(a):=[1/2](['(s)]^2a^+-s^2a^-).$
An interesting situation is when $j$ is a convex function, $[\^\,(\mathbf{E})][j(X)]=E[j(X_0)]$ with $X_0 \sim N(0, ['(s)]^2)$. But if $j$ is a concave function, then the above $['(s)]^2$ has to be replaced by $s^2$. This coincidence can be used to explain a well-known puzzle: many practitioners, particularly in finance, use normal distributions with `dirty' data, and often with successes. In fact, this is also a high risky operation if the reasoning is not fully understood. If $s=['(s)]=s$, then $N(0, [s^2, ['(s)]^2])=N(0, s^2)$ which is a classical normal distribution. The method of the proof is very different from the classical one and a very deep regularity estimate of fully nonlinear PDE plays a crucial role.
A type of combination of LLN and CLT which converges in law to a more general $N([m, ['(m)]], [s^2, ['(s)]^2])$-distributions have been obtained. We also present our systematical research on the continuous-time counterpart of the above `G-normal distribution', called G-Brownian motion and the corresponding stochastic calculus of Itô's type as well as its applications.
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