I am reading a paper and wanted to check if the following random variables are identically distributed.
Let $X_1,X_2,\cdots,X_n$ be iid and $X_j\sim \mu$ for all $j\in\{1,\cdots,n\}$ where $\mu$ is some distribution. Let $Z=f(X_1,X_2,\cdots,X_j,\cdots, X_n)$ and $Z_j=f(X_1,X_2,\cdots,X_j',\cdots,X_n)$ where $X_j'\perp \!\!\! \perp X_j, X_j'\sim \mu$ and
\begin{align}
f(x_1,\cdots,x_n) &= P(\sum_{i=1}^n X_i=\sum_{i=1}^n x_i) \\
f_j(x_1,\cdots,x_n) &= P(\sum_{i\neq j} X_i=\sum_{i\neq j} x_i)
\end{align}
It is given in the paper that $Z-f_j(X_1,\cdots,X_n)$ and $Z_j-f_j(X_1,\cdots,X_n)$ are identically distributed because $X_j'$ is an independent copy of $X_j$ and $f_j$ does not depend on the $j^{th}$ variable.
Q1) Is it true that $Z$ and $Z_j$ are identically distributed?
Q2) If the above is true, then $Z-f_j$ and $Z_j-f_j$ are also identically distributed. So may I know where we use the fact that $f_j$ does not depend on the $j^{th}$ variable?
Best Answer
It's false as you state it. Instead of the condition $X'_j\perp X_j$, you need that $X'_j\perp(X_i, i\ne j)$, so that $X_1, X_2, \dots, X'_j, \dots, X_n$ is also an i.i.d. collection.
Under that condition, the vectors $(X_1, \dots, X_j, \dots, X_n)$ and $(X_1, \dots, X'_j, \dots, X_n)$ have the same distribution. So if $g$ is any function, then $W:=g(X_1, \dots, X_j, \dots, X_n)$ and $W':=g(X_1, \dots, X'_j, \dots, X_n)$ have the same distribution.
Now if you've got some further thing that "does not depend on the $j$th variable", say $Y=h(X_1, \dots, X_{j-1}, X_{j+1}, \dots, X_n)$, then $W-Y$ and $W'-Y$ have the same distribution. This is just another application of the previous paragraph, since $W-Y$ is a function of $X_1,\dots,X_j,\dots, X_n$ and $W'-Y$ is "the same function" of $X_1,\dots, X'_j,\dots, X_n$.
However, if you had a second quantity that did depend on the $j$th coordinate, say $Z=r(X_1, \dots, X_j, \dots, X_n)$, then you couldn't conclude that $W-Z$ and $W'-Z$ have the same distribution. In that case $W'-Z$ would be a function of all $n+1$ variables $X_1,\dots,X_j, X'_j, \dots, X_n$.