With Small-Ball Probability (SmBP) we mean the behavior of the probability that a random process X is contained in a ball centered in a fixed point and whose radius tends to zero. A classical hypothesis assumes that the SmBP can be factorized into two terms that isolate the spatial contribution (i.e. the dependence on the center of the ball) and the volumetric contribution (i.e. the dependence on the radius) respectively. The first factor plays the role of pseudo-density for X and allows to define some classification methodologies. The second factor allows to evaluate the complexity of the underlying process X and can be used in building some goodness-of-fit test procedures.
