Average over spheres finite

Question

Let $X_1,...,X_N$ be random variables that are iid with the uniform distribution over $\mathbb S^n.$

I am curious how to see that $f(X_1,..,X_N):=\left \lvert \sum_{i=1}^N X_i \right\rvert^{-1}$ has finite expectation, i.e.

$$\int_{(\mathbb S^n)^N} f(X_1,..,X_N) \ dS(X_1)\cdots dS(X_N)<\infty$$

is finite, where $dS$ is the surface measure.

I know it is true by this answer here, which shows it for $\mathbb S^1$ and gives even the asymptotic of the integral. So I believe that if it is true on $\mathbb S^1$ it also has to be true on all other spheres, but I am looking for a more direct argument than in the above answer to see that this is indeed the case.

Answer 1

Assuming you mean $f(X_1,\ldots,X_n) = |\sum_{i=1}^N X_i|^{-1}$, the case $n\geq 2$ is actually easier than $n=1$. You could use that there exists a constant $C>0$ such that for any $x\in\mathbb{R}^n$ the expectation $\mathbb{E}[ 1 / |X_1 + x| ] < C$. Then automatically also $\mathbb{E}[ 1 / |X_1 + \ldots + X_n| ] < C$ by taking $x$ to have the distribution of $X_2 + \ldots + X_N$.

Note that this argument does not work for $n=1$, because if $|x|=1$ then $\mathbb{E}[ 1 / |X_1 + x| ] = \infty$ due to a logarithmic divergence.