The $N^{-1}(G_x(U_x))$ are standard normal, and $M_n$ is standard multivariate normal. This is because when $G_x(U_x)$ is uniform, the inverse of the cumulative distribution function of a standard normal distribution, denoted as $N^{-1}$, applied to $G_x(U_x)$ results in a standard normal distribution. This is due to the property of the Gaussian copula, which transforms uniformly distributed random variables into standard normally distributed random variables. On the other hand, $M_n$ represents a standard multivariate normal distribution. This is because in the context of the Gaussian copula, when multiple variables are involved (n-variate case), the joint distribution of these variables follows a standard multivariate normal distribution. This is a fundamental property of the Gaussian copula, which allows it to capture the dependence structure among multiple variables. Therefore, when $G_x(U_x)$ is uniform, both $N^{-1}(G_x(U_x))$ and $M_n$ follow their respective standard normal and standard multivariate normal distributions.

Choice A is incorrect. $M_n$ are standard multivariate normal, while $N^{-1}(G_x(U_x))$ are univariate normal.

Choice B is incorrect. This choice incorrectly states that $N^{-1}(G_x(U_x))$ are multivariate normal and $M_n$ is univariate normal. The correct characterization of these terms should be: $N^{-1}(G_x(U_x))$ are standard normal and $M_n$ is standard multivariate normal.

Choice D is incorrect. This choice incorrectly states that both $N^{-1}(G_x(U_x))$ and $M_n$ are standard normal. While it's true for the former, the latter ($M_n$) should be characterized as a standard multivariate normal distribution.

Things to Remember

• Gaussian copula is a popular choice in finance due to its ability to model the dependence structure between multiple variables.
• Uniform distribution is often used as a starting point in copula modeling, as it can be transformed into other distributions like the standard normal distribution.
• The inverse of the cumulative distribution function (CDF) of a standard normal distribution is denoted as $N^{-1}$ and is used to transform variables to a standard normal distribution.
• In the context of copulas, multivariate normal distributions are used to represent the joint distribution of multiple variables.