Independence of shifted random variables

Question

Suppose that

$(X,Y_{1},...,Y_{N})$

is a collection of mutually independent random variables. Is it true that

$(X-Y_{1}, X-Y_{2},...,X-Y_{N})$

are mutually independent? I can't seem to find any references on this.

Answer 1

No, it is not true. Let $(X,Y_1,\dots,Y_N)$ be i.i.d. Bernoulli$(1/2)$, so each is equal to either $0$ or $1$ with equal probability. I will let $P(A,B,C)$ refer to $P(A\cap B\cap C)$, for short. $$ P(X-Y_1=1,X-Y_2=1)=P(X=1,Y_1=0,Y_2=0)=1/8, $$ but \begin{align} P(X-Y_1=1)P(X-Y_2=1) &=P(X=1,Y_1=0)P(X=1,Y_2=0) \\&=(1/4)(1/4) \\&=1/16. \end{align}

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Edit: Even more simply, if $Y_1,Y_2,\dots,Y_N$ are all zero with probability $1$, then $(X-Y_1,\dots,X-Y_N)=(X,X,\dots,X)$ is not independent, because $X$ is not independent of itself (as long as $X$ is non-constant).