Is Earth Mover Distance has maximum bound? I have two probability distributions which each distribution has sum up to 1. 
I want to compute the distance between those two probability distributions. I want to use Earth Mover Distance to calculate the distance between those two distributions. 
What I have found so far is the following, that $EMD(p_X,q_X)$ between distributions $p_X$ and $q_X$ is:
$$EMD(P,Q) = \frac{\sum^m_{i=1}\sum^n_{j=1} f_{ij}d_{ij}}{\sum^m_{i=1}\sum^b_{j=1} f_{ij}}$$
my question is, is there anyone has a mathematically proof that EMD is bounded distance. While two probability distributions have the same condition, sum up to 1. What's is the maximum possible value of the EMD that can be achieved (maximum bound)? 
 A: The answer to your question as stated is no, unless your two random variables happens to be defined on a finite (or compact) metric space. But for real (or integer ...) valued random variables, certainly not. Detailed answers can be found from here:  Earth Mover's Distance (EMD) between two Gaussians  which analyzes (and lower-bounds) earth mover distance between two normal distributions.
It could also be easily answered from the interpretation "Earth Mover's Distance can be formulated and solved as a transportation problem. Suppose that several suppliers, each with a given amount of goods, are required to supply several consumers, each with a given limited capacity. For each supplier-consumer pair, the cost of transporting a single unit of goods is given. The transportation problem is then to find a least-expensive flow of goods from the suppliers to the consumers that satisfies the consumers' demand." from https://en.wikipedia.org/wiki/Earth_mover%27s_distance. Just move supplier and consumer at arbitrarily large distance ... Concretely, if both supplier and consumer lives on the surface of planet earth, an upperbound would be about 20000 km (that is an example of the compact case).
For information on such questions, a good source of information is in https://www.amazon.com/Encyclopedia-Distances-Michel-Marie-Deza/dp/3642309577
