# probability – maps a random variable to a Gaussian

If I have a random variable $$X in mathbb {R} ^ n$$, under what conditions there is a $$C ^ 1$$ function $$varphi: mathbb {R} ^ n rightarrow mathbb {R} ^ n$$ such that $$varphi (X) sim mathcal {N} (0, I_n)$$ (vector $$n$$ independent normal variables)?

$$X$$ It probably has to have a density to begin with … But I'm a bit stuck there. I know that if $$n = 1$$, the inverse transformation sampling can be used to solve this problem, but I have trouble finding a generalization to $$R ^ n$$.