Find, with proof, an example of a distribution function $F$ in $\mathbb{R}^d$ with absolutely continuous margins or component functions that is not absolutely continuous.
I know that if $F$ is absolutely continuous, its margins are absolutely continuous. Also, $F$ is absolutely continuous if it can be written in the form $\int_{-\infty}^{x_1} \cdots \int_{-\infty}^{x_d} f(z_1,\cdots, z_d) dz_1\cdots dz_d,$ where $f$ is the density of $f$. I also know that random vectors are vectors of random variables. However, I'm not sure how to find a distribution function $F$ with absolutely continuous margins that is not absolutely continuous.
Best Answer
Let $X$ have a standard normal distribution and $X_i=X$ for $1\leq i \leq n$. Then the distribution function of $(X_1,X_2,...,X_n)$ is concentrated on the set $\{(x,x,...,x):x \in \mathbb R\}$ which has Lebesgue measure $0$ so it is not absolutely continuous. But the marginals are normal.