Let X1 , . . . be independent with common mean μ and common variance σ 2 , and set $\displaystyle Y_n = X_n + X_{n+1} + X_{n+2}$ . For $\displaystyle j \leq 0$, find $\displaystyle Cov(Yn , Yn+j)$.

If$\displaystyle j=0$ then $\displaystyle Cov(Y_n,Y_n)=Cov(Y_n)=3\sigma^2$

If $\displaystyle j=1$ then $\displaystyle Cov(Y_n,Y_{n+1})=Cov(X_n+X_{n+1}+X_{n+2},X_{n+1}+X _{n+1+1}+X_{n+2+1})$ I don't see how I can do anything with this statement.

or I want to just apply the definition of covariance, $\displaystyle Cov(X,Y)=E[(X-E[X])(Y-E[Y])]$, to get $\displaystyle Cov(Y_n, Y_{n+1})=E[Y_n*Y_{n+1}-Y_{n+1}*E[Y_n]-Y_n*E[Y_{n+1}]+E[Y_{n+1}]*E[Y_n]]$, which seems too complicated to be helpful.