Timeline for When $(X_1-X_0, X_1-X_2)\sim (X_2-X_0, X_2-X_1)\sim(X_0-X_1, X_0-X_2)$?

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Oct 9, 2020 at 13:10	history	edited	Dilip Sarwate	CC BY-SA 4.0	added 53 characters in body
Dec 12, 2018 at 13:33	comment	added	Dilip Sarwate		The marginal distributions of $X$ and $Y$ are completely determined by the joint distribution of $X$ and $Y$. a.k.a. the distribution of $(X,Y)$. So, if you assume $(A,B) \sim (X.Y)$, you are implicitly assuming that $A\sim X$ and $B\sim Y$. Thus, your transitive assumption that $(X_1,X_1-X_2)\sim (-X_1, -X_2)$ is saying that $X_1 \sim -X_1$ which is your symmetry condition on $G_1$ but also that $X_1 - X_2 \sim -X_2$ which can only happen if $X_1$ is $0$ with probability $1$. Note that your $X_1-X_2 \sim -X_2$ also implies that.$X_1=0$. var($X_1-X_2)=2$, var$(-X_2)=1$ in your example.
Dec 12, 2018 at 8:55	comment	added	Star		To avoid confusion I've slightly modify my question, adding $X_0$.
Dec 11, 2018 at 21:39	history	answered	Dilip Sarwate	CC BY-SA 4.0