Are different t distributions generated by the same set of iid RVs independent?

Question

Suppose $X_0,...X_n$ iid $\mathcal{L}(X_0)=\mathcal{N}(0,1)$.

Define the t-distributions w/ n degrees of freedom: $$Z_i:=\frac{\sqrt{n}X_i}{\sqrt{\sum_{\underset{j\neq i}{j=0}}^nX_j^2}}$$ Are $\{Z_0,..,Z_n\}$ always independent? I would like to know a proof or in this case probably a counterexample. You are free to comment yes/no.

($n\geq 2$)

Answer 1

You can already see that the issue of the denominator excluding the term $X_i^2$ is not an impediment to showing dependence, since with a minor modification, $$Z_i^{-2} + \frac{1}{n} = \frac{\sum_{j=0}^n X_j^2}{n X_i^2}$$ implies $$\sum_{i=0}^n \left( Z_i^{-2} + \frac{1}{n} \right)^{-1} = n.$$

Answer 2

Dependency: $\sum_{i=0}^n Z_i^2=n$ This shows the $Z_i$ cannot be independent.
A simple example $(n=1)$ then $Z_0^2+Z_1^2=1$.