Concentration order on a metric space, with some statistical applications

A concept called concentration order of probability distributions on a metric space is introduced, then the norm stochastic order that compares real-parameter stochastic processes is introduced as a special case. Equivalent conditions of concentration order are established. Using Anderson's theorem, norm stochastic orders of certain processes with almost-sure continuous sample paths are established, including Wiener processes and Brownian bridges as special cases. Limiting power monotonicity of some goodness-of-fit tests are shown as applications.