Say I have two i.i.d. samples X1,X2,...,Xn and Y1,Y2,...,Yn with population means E(X) and E(Y), and population variance var(X) and var(Y). If I apply the CLT, then the scaled (by square root n) and centred (by the population mean) sample means of X and Y are distributed N(0, Var(X)) and N(0, Var(Y)).
I know that any function of X or Y, f(Xi) or f(Yi), is also i.i.d so I can apply the CLT again. But what about functions of both of them f(Xi,Yi). In particular f(Xi,Yi) = Xi + Yi and f(Xi,Yi) = Xi * Yi. Can I still apply the CLT? Can I only apply it in certain circumstances (like if I assume the two samples are also independent of each other)?
Thanks very much,
Daniel