Partition-Based Probability Measures for Discontinuous and Mixed Distributions

Abstract Conventional parametric probability models provide parsimonious representations of probability laws but may be inadequate when the underlying distribution exhibits abrupt changes in its density, multiple components, isolated atoms, or a combination of discrete and continuous probability. This paper develops a partition-based framework for constructing and estimating probability measures without requiring a single prescribed functional form for the entire distribution. The construction assigns probability masses to measurable cells and permits discrete, continuous, and mixed conditional distributions within the cells. We investigate the approximation of arbitrary probability measures by partition-induced discrete measures, establish weak convergence under partition refinement, and derive statistical consistency for estimated partition distributions. Particular attention is given to the distinction between weak convergence and uniform convergence of cumulative distribution functions when the target distribution has atoms. An atom-preserving approximation scheme is introduced for which uniform CDF approximation is controlled by the maximum probability of the cells used to approximate the nonatomic component. We further establish finite-partition asymptotic normality, derive a Wasserstein error bound, and formulate conditions for consistency when partition boundaries are estimated. A simulation design is proposed to investigate the finite-sample behaviour of the construction under continuous, discontinuous-density, discrete, and mixed probability laws. The results provide a unified mathematical basis for studying partition-based approximation and statistical inference for probability distributions with irregular structure. Mathematics Subject Classification (2020): 60A10; 60B10; 62G05; 62G20.