I read the book 'Numerical Methods for Stochastic Computations: A Spectral Method Approach' by Dongbin Xiu, Chapter 4. The main goal of the problem is, given random variables $Y_1, ...,Y_n$ (on a probability space, say, $\Omega$), find independent random variables $Z_1,...,Z_d, 1\le d\le n$, so that $Y=T(Z)$ for some transformation function $T$.
Note that two random variables $Y_1, Y_2$ are independent if $P(Y_1\in B_1 \cap Y_2\in B_2)=P(Y_1\in B_1)P(Y_2\in B_2)$ for any measurable sets $B_1, B_2$, where $P$ is a probability measure (Don't be confused with linear independence).
Anyway, my problem is when $n=2$. If $Y_1, Y_2$ are independent, we can take $Z_i=Y_i$.
When $Y_1$ and $Y_2$ are not independent, the book says that there exists a (nonzero) function $f$ such that $$f(Y_1,Y_2)=0$$ so that we can find a random variable $Z(\omega)$ to parametrize $Y_1=a_1\circ Z, Y_2=a_2\circ Z$ and $f(a_1,a_2)=0$. Or equivalently, we may find a function $g$ so that $Y_2=g(Y_1)$, so that $Z=Y_1, Y_2=g\circ Z$. (So we found one variable $Z$, which is cleary satisfies the independent statement, and a transfrom $a_i$ or $g$)
First, I don't know how to find such functions $f, g, a, b$ and second, I don't know why those two results are equivalent. Any references will be appreciated!