Title: A generalized probability kernel on discrete distributions and its application in two-sample test. Abstract: We propose a generalized probability kernel(GPK) on discrete distributions with finite support. This probability kernel, defined as kernel between distributions instead of samples, generalizes the existing discrepancy statistics such as maximum mean discrepancy(MMD) as well as probability product kernels, and extends to more general cases. For both existing and newly proposed statistics, we estimate them through empirical frequency and illustrate the strategy to analyze the resulting bias and convergence bounds. We further propose power-MMD, a natural extension of MMD in the framework of GPK, illustrating its usage for the task of two-sample test. Our work connects the fields of discrete distribution-property estimation and kernel-based hypothesis test, which might shed light on more new possibilities.