I'm interested in some instances of the following problem.
Let $n \geq 2$, and suppose we draw $k \geq 2$ vectors $v_1, \dots, v_k$ uniformly at random from the $n$-dimensional ball of radius $1$, $B_n(1)$. Consider the vector $w = v_1 + \dots + v_k$. What is the probability $p(n,k,d)$ that this vector $w$ has norm at most $d$?
For $n = 2$ and $k = 2$, the answer can be obtained by computing the area of the intersection of two circles, and relating it to the area of the whole circle. Similarly, for $n = 3$ and $k = 2$ some results are known to compute the probability exactly. However, I'm mostly interested in larger $n$ and, if possible, slightly varying $k = 2, 3, \dots$ and $d = 1 - \epsilon$ for small $\epsilon > 0$.
My main questions are: Is this general problem known in literature? Does it have a specific name? What is known about higher-dimensional cases?
For $k = 2$, I already do have some reasonable lower and upper bounds on $p(n,k,d)$ for large $n$ which are "only" a polynomial factor (in $n$) apart, and the asymptotic behavior seems pretty clear. But already for $k = 3$, doing anything analytically seems very hard, even when $d = 1$. Any references or help in approaching this problem would be appreciated.
Thanks in advance.
Edit: To clarify, I am mostly interested in asymptotics for reasonably large $n$, while $k$ remains small.