Jan 9: Copula and Sklar Theorem

当构建多个随机变量的联合分布时，可以分两步，第一先有每个变量的边际分布。第二描述这些变量如何related to each other，这个描述就是Copula。精确定义如下

A copula is the joint distribution of random variables U1, U2, . . . ,Up, each of which is marginally uniformly distributed as U(0, 1).

所以Copula其实是个多元函数。Sklar定理保证这种描述方式make sense。因为此多元函数存在并且在连续情况下唯一。

两种常见Copula：Gaussian Copula和t-Copula。

When R = I, the multivariate normal distribution is that of independent standard normal variables, and the copula has a constant density. But the t-copula still shows dependence.

Tail Dependence, which is important in risk management, is coming tomorrow.