Independence of discrete random variable

Question

Let $U_i$,$V_i,\ i=1,\dots,n$ be $2n$ real random variables i.i.d..

Are $(U_i-V_i)^2,\ i=1,\dots, n$ independent ?

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I should check that

$$P((U_i-V_i)^2=k,(U_i-V_i)^2=p)=P((U_i-V_i)^2=k)P((U_j-V_j)^2=p)$$

is there any general theorem about such independence or should I use generating function to make the computation ?

Answer 1

I think theorem 2.1.10 of Rick Durret's Probability: Theory and Examples is what you are looking for.