Comonotonic and Countermonotonic RV's and their relation to Frechet Hoeffding Bounds If $F_1 ... F_d$ are are all continuous, and $X_j \sim F_j$, $j= 1...d$, then the Frechet upper bound corresponds to comonotonic random variables with 
$$ X_j = F^{-1}_j(F_1(X_1)) \ \ \ \ \ \ \ \ [1] $$
In literature, I have seen that comonotonicity corresponds to the Frechet Hoeffding upper bound, but I do not understand why.  The upper bound can be written as:
$$ F^+(\mathbf{x}) = min(F_1(x_1) ... F_d(x_d)) \ \ \ \ \ \ \ \ [2] $$
I am not looking for a rigorous proof, although that might also be helpful, but can someone explain why Equation 1 above corresponds to Equation 2?  An intuitive explanation might be helpful too.
 A: The first equation essentially means that the rank correlation between $X_i$ and $X_j$ is one for $i \neq j$.  Put another way, whenever you sample from this joint distribution the individual elements of $X = (X_1, X_2, \ldots , X_d)$ will almost surely correspond to the same quantiles from their marginal distributions.
Now let's think about the joint distribution function: $F(x) = P(X_1 \leq x_1 \cap X_2 \leq x_2 \cap \ldots \cap X_d \leq x_d)$, and take $i^\prime$ to be the index for which $F_{i^\prime}(x_{i^\prime})$ is the smallest.  This means that $x_{i^\prime}$ is the smallest quantile out of $x_1, x_2, \ldots, x_d$ with regard to the marginal distributions $F_1, F_2, \ldots, F_d$.  Because of comonotonicity the event $\{ X_{i^\prime} \leq x_{i^\prime} \}$ almost surely coincides with the event $\{ X_1 \leq x_1 \cap X_2 \leq x_2 \cap \ldots \cap X_d \leq x_d \}$ because if $X_{i^\prime} \leq x_{i^\prime}$, then every other event in this set also occurs with probability one, and the latter event of course implies the former.  This means the two events have the same probability, and so you get equation two.
