Sum of independent random variables and absolute continuity. Helly Everyone!
There are two questions that have been burning in my mind in the last couple of days and I would really appreciate if somebody could help me with them.
Let us start by assuming a probability space $(\Omega, \mathcal{F}, P)$ and $\{ X_1, ..., X_J \}$, i.e. a collection of $J$ independent random variables. My questions are these:

*

*If we take any random variable from the original collection, lets say $X_i$, is the new collection $\mathcal{A} = \{ X_1 - X_i, ...,X_{i - 1} - X_i,\; X_{i + 1} - X_i, ... , X_J - X_i \}$ independent?


*Now, assume that the vector $(X_1, ..., X_J)$ (which includes $X_i$) is absolutely continuous w.r.t. the Lebesgue measure on $\mathcal{B}(\mathbb{R}^J)$ (denoted here $\lambda^J$), how can we prove that the vector $(X_1 - X_i, ...,X_{i - 1} - X_i,\; X_{i + 1} - X_i, ... , X_J - X_i)$ is absolutely continuous w.r.t. $\lambda^{J - 1}$?
A few things worth noting: It is relatively simple to prove that if $\{ X_1, ..., X_J \}$ is independent, than $\{ X_1, X_{i - 1}, - X_i, X_{i + 1}, ..., X_J \}$ is also independent, so the distribution of each of the subtractions is a convolution. When it comes to the first question, this is basically where I ran out of ideas.
For the second one, assuming that the collection of subtractions is independent, we can prove that any measurable rectangle of measure zero according to Lebesgue also has measure zero according to the distribution of $(X_1 - X_i, ...,X_{i - 1} - X_i,\; X_{i + 1} - X_i, ... , X_J - X_i)$. My idea was to follow that up with some application of the $\pi - \lambda$ theorem, but I just couldn't think of the set.
Any help is greatly appreciated!
 A: It should intuitively be clear that the set $\mathcal A$ is not a set of independent variables, as they all involve the common summand $X_i$. 
As already pointed out, you can prove this by calculating their covariances $$\mathbb E\big[(X_k-X_i-\mathbb E[X_k-X_i])(X_j-X_i-\mathbb E[X_j-X_i])\big]$$ If you know the relationship between (in-)dependence of random variables and their covariances you should be able to get the answer.
Now regarding the second question:
Suppose $A\subset \mathbb R^{J-1}$ is a $\lambda^{J-1}$-null-set. Define the set $$B=\{x\in \mathbb R^J: (x_1-x_i,\dots,x_{i-1}-x_i,x_{i+1}-x_i,\dots,x_J-x_i)\in A\} \subseteq \mathbb R^J$$ Think of whether $B$ must necessarily be a $\lambda^J$-null-set.
If that's the case, it would follow that $\mathbb P[(X_1,\dots,X_J)\in B]=0$ and hence $\mathbb P[(X_1-X_i,\dots,X_{i-1}-X_i,X_{i+1}-X_i,\dots,X_J-X_i)\in A]=0$.
Edit:
I'm going to elaborate more on what i intended with the second hint. For each $M>0$ define $$B_M:=\{x\in B: |x_i|\leq M\}$$ Now since $\lambda^{J-1}$ is translation invariant we have by $\lambda^J=\lambda^{J-1}\otimes\lambda$ that $$\lambda^J(B_M)\leq \lambda^{J-1}(A)\times \lambda([-M,M])$$
Now note that $$B=\bigcup_{M=1}^\infty B_M$$
