In page 45 from the book Numerical Methods for Stochastic Computations, by Dongbin Xiu, it is stated that, since $\alpha$ and $\beta$ are not independent random variables, then there exists a function $f$ such that $f(\alpha,\beta)=0$. Furthermore, it says that, equivalently, the dependence between $\alpha$ and $\beta$ implies that there exists a function $g$ such that $\beta=g(\alpha)$. Why does non-independence imply the existence of $f$ and/or $g$ ?
Edit: Dependent means not independent. The function $f$ is nonzero.
Without additional assumptions, neither the existence of a nonzero function $f$ nor the function $g$ follows.
For a simple example, pick a point on the unit disk with uniform probability and let α and β be the $x$ and $y$ coordinates of the chosen point. Clearly dependent, and just as clearly neither is a function of the other. Furthermore, the only way for $f(\alpha,\beta)$ to be zero is for it to vanish uniformly.