# What is this probability distribution?

Thank you in advance for any suggestions or feedback.

I have a discrete 1D probability distribution represented as a vector $$\textbf{p}$$, $$p_i = p(x_i)$$.

I am interested in finding the Wasserstein (earth mover's) distance between $$\textbf{p}$$ and a random permutation of itself, $$\textbf{p}'$$. i.e.

$$W = \sum_{i=0} \mid \sum_{j=0}^i p_j - \sum_{j=0}^i p_j' \mid$$

My question is what is the probability distribution of $$W$$, given $$\textbf{p}$$? It's of course positive and, from simulations, heavy near the origin. So maybe a gamma or beta distribution? It would be amazing to see a way to derive this. Any guidance would be greatly appreciated.

• The distribution of $W$ is discrete, bounded, and heavily influenced by the specific probabilities, suggesting that continuous models like Gamma or Beta are unlikely to work well. Is there anything more you can say about the probabilities themselves? What do you perceive to be the limitation of the simulation results? – whuber Feb 15 at 17:09
• Thank you very much whuber. I am looking to calculate a p-value, null hypoth being that p is randomly permuted, but bootstrapping is hard because the distribution is very long tailed. So I am looking for a way to get the analytical shape of the distribution, then use simulation to find the parameters of the distribution. – user979797987678 Feb 15 at 17:17
• @whuber to clarify, p is given. i.e. this is not asking for a distribution over different p – user979797987678 Feb 15 at 17:36
• Understood: but the distribution of $W$ depends on the given $p.$ – whuber Feb 15 at 19:30