so i got the following problem:
consider the follow.
$$\overline{C}_{\overline{X}}:\mathbb{R}\to [0,1], u\mapsto\overline{C}_{\overline{X}}(u) = \mathbb{P}(\overline{F}_{X_1}(X_1)\leq u_1,...,\overline{F}_{X_d}(X_d)\leq u_d)$$
defines a copula and $\overline{X} = (X_1,...,X_d)$ is a vector of continuous random variables
my solution so far
first recall that a copula is a joint distribution function whose marginal distributions are all uniform on $[0,1]$. Thus, if we can show that $\overline{C}_{\overline{X}}$ is the joint cdf of a random vector with uniform marginals, then it follows that $\overline{C}_{\overline{X}}$ is a copula.
To show that $\overline{C}_{\overline{X}}$ is the joint cdf of a random vector with uniform marginals, we can define a new random vector $\mathbf{U}=(U_1,\ldots,U_d)$ with uniform marginal distributions, i.e., $U_i\sim U(0,1)$ for $i=1,\ldots,d$. Then, we can define a new random vector $\mathbf{V}=(V_1,\ldots,V_d)$ as $V_i=\overline{F}_{X_i}(X_i)$ for $i=1,\ldots,d$.
first**(unsure if this is needed or if there is something else i need to do, maybe the tail ??)** we need to show that, $\overline{F}_{X_i}(X_i)\sim U(0,1)$ for all $i$. we will do this by using the probability integral transformation thm, which states that
\begin{align*}
\mathbb{P}(V_i\leq v) & = \mathbb{P}(F_{X_i}(X_i)\leq v)\\
& \overset{1}{=} \mathbb{P}(X_i\leq F_{X_i}^{-1}(v))\\
& \overset{2}{=} F_{X_i}(F_{X_i}^{-1}(v))\\
& \overset{3}{=} v
\end{align*}
\
- I am using the definition of V. 2) I am using the fact that $X_i$ has cdf $F_{X_i}$ \
thus we can conclude $V_i$ has a uniform distribution on the interval from 0 to 1. and therefore V has uniform marginal\
Now, we can show that the joint cdf of $\mathbf{V}$ is equal to $\overline{C}_{\overline{X}}$ as follows: \begin{align*} F_V(v_1,...,v_d) & \overset{1}{=} \mathbb{P}(V_1\leq v_1,...,V_d\leq v_d)\\ & \overset{2}{=} \mathbb{P}(\overline{F}_{X_1}\leq v_1,...,\overline{F}_{X_d}\leq v_d)\\ & \overset{3}{=} \overline{C}_{\overline{X}}(v_1,...,v_d) \end{align*}
1 follows from the definition of joint cdf. 2 follows from the definition of $\mathbf{V}$, and 3 follows from the definition of $\overline{C}_{\overline{X}}$.\ \ Therefore, $\overline{C}_{\overline{X}}$ is the joint cdf of a random vector with uniform marginals, and hence it defines a copula.\
wrote a question in the text, so i could be in context with the rest. Any thing else is appreciated and welcome.
Thanks in advance