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Suppose $X_1, \ldots, X_4$ are random variables such that $X_1$ is perfectly correlated with $X_2, X_3, X_4$, such that

$$ Corr(X_i, X_j) = 1 \qquad \forall i\neq j $$.

I am wondering if $X_1$ is also perfectly correlated with $X_2+X_3+X_4$. In general, if $X_1$ is perfectly correlated with any $n$ variables, is $X_1$ independent of $\sum_{i=2}^nX_i$ as well? Thanks.

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Hint:

In general if $X,Y$ are non-degenerate random variables that have variances then: $$\mathsf{Corr}(X,Y)=1\iff\mathsf{Var}(\sigma_XY-\sigma_YX)=0\iff\sigma_XY-\sigma_YX\text{ is degenerated}$$

That comes to the same as the existence of reals $a>1,b$ with $X=aY+b$ a.s.

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Yes, of course..See that $r_{X_1,X_2}=1 \implies X_2=a_1 X_1+b_1, r_{X_1,X_3}=1 \implies X_3=a_2 X_1+b_2,r_{X_1,X_4}=1 \implies X_4=a_3 X_1+b_3$ where $a_i's $ and $b_i s$ are constants. Thus $X_2+X_3+X_4=(a_1+a_2+a_3)X_1 + (b_1+b_2+b_3)$ which means that they are perfectly correlated. See that $a_1+a_2+a_3$ cannot be $0$.

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