Let $X_0$,$X_1$,....,$X_n$ denote i.i.d. real valued r.v.'s.
For every definition of $Y_1$,...,$Y_n$ below, say whether or not $Y_1$,...,$Y_n$ are exchangeable and justify your answer.
- $Y_j$ = $X_j$ - $X_0$, j = 1,...,n
- $Y_j$ = $X_j$ - $X_{j-1}$ , j = 1,...,n
- $Y_j$ = $X_j$ - $\bar{X}$, j = 1,...,n where $\bar{X}$ = $\sum\limits_{j=1}^n X_j/n $
- $Y_j$ = (j/n)$X_j$ + (1 - j/n)$X_0$, j = 1,...,n
I. Yes, since each $X_j$ is subtracting the same value, and since each $X_j$ is i.i.d., $Y_j$ is exchangeable.
II. No, since each $X_j$ depends upon the previous $X_{j-1}$.
III. Yes, each $X_j$ is simply subtracting the mean of $X_j$'s. This should have no effect on their exchangeability.
IV. Not sure.