Is the sum or product of two iid random process still a iid process? For example, let $X_n $ is iid Gaussian random process, and $U_n $ is iid binary random process, and they are independent to each other.

$Z_n=X_n U_n$;$Y_n=X_n +U_n$;$R_n=X_n + U_0$

Are $Z_n$,$Y_n$ and $R_n$ still iid processes?


closed as off-topic by Math1000, jvdhooft, Ethan Bolker, José Carlos Santos, The Phenotype Jun 12 '18 at 20:24

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  • $\begingroup$ For example $X,Y$ independent implies that $X+Y$ is independent: math.stackexchange.com/questions/1546247/… $\endgroup$ – Math1000 Jun 12 '18 at 10:11
  • $\begingroup$ but they are variable,mine is process $\endgroup$ – XM551 Jun 12 '18 at 10:14
  • $\begingroup$ Surely, yes. $Z_n,Y_n,R_N$ are all i.i.d.. $\endgroup$ – Kabo Murphy Jun 12 '18 at 10:19
  • $\begingroup$ can you give me some proof? $\endgroup$ – XM551 Jun 12 '18 at 10:21
  • $\begingroup$ A stochastic process is a collection of random variables. In the case where they are indexed on the positive integers, it follows clearly from induction that if e.g. $X_1+U_1$ are independent then so too are $X_{n+1}+U_{n+1}$ for all $n$. $\endgroup$ – Math1000 Jun 12 '18 at 10:30