Cfrgtkky

Say I have some arbitrary $N$-dimensional probability distribution $P$.

I'd like to create a deterministic function that accepts a vector of $N$ uniformly distributed numbers and generates an $N$ element vector where the elements of this vector are distributed according to $P$.

Is this always possible?

1. If $N=1$ then it is always possible.
   Let $$ Xsim F, $$
   there exists $$F^{-1}(u)=inf {xin mathbb{R}colon F(x)ge u}$$ where $0< u<1$. Thus $$ F^{-1}(U)sim F$$
   with $Usim Unif(0,1)$. This method is called inverse transform.




2. If $N>1$. Let $$ X=(X_1,X_2,dots,X_N)sim F $$
   where $$F(x_1,dots,x_N)=mathbb{P}(X_1le x_1,dots, X_Nle x_N).$$
   If $X_1,dots,X_N$ are independent, then $$F(x_1,dots,x_N)=mathbb{P}(X_1le x_1)cdots mathbb{P}(X_Nle x_N)=F_{X_1}(x_1)cdots F_{X_N}(x_N).$$
   Thus is possible to simulate each $X_i$ by inverse transform method and so
   $$ (F_{X_1}^{-1}(U_1),dots,F_{X_N}^{-1}(U_N))sim F$$
   where $U_1,dots,U_N$ i.i.d. with $U_1sim Unif(0,1)$.
   If $X_1,dots,X_N$ are dependent then a procedure that takes this into account is necessary.