Distribution in Unit disk

Question

Let $X$ and $Y$ be i.i.d random variables that are uniformly distributed on the positive orthant of the unit sphere in $\mathbb{R}^n$. What is the distribution of $X - Y$? In particular, does $X-Y$ yield a uniform distribution over the angles of the vectors obtained by subtracting two uniformly random vectors on the positive orthant of the unit sphere?

Answer 1

The distribution isn't spherically symmetric. The probability of $X-Y$ to end in all-positive or all-negative orthant is zero, since there is no such two points $x,y$ on an orthant that $x_i>y_i$ for all $i$.

If you have meant the interior of unit sphere in the positive orthant, then it's less obvious, but the answer is the same. you have non-zero probability density near $(1,-1,0,0\ldots0)$ and zero probabaility density near $(1,1,0,\ldots0)$