Superpositions of Probability Distributions

TL;DR

This paper explores superpositions of Gaussian probability distributions with varying variances, revealing their mathematical properties, applications in simplifying complex equations, and extensions to quantum mechanics.

Contribution

It derives the form of smearing distributions that preserve the semigroup property and demonstrates their applications in probability theory, path integrals, and quantum mechanics.

Findings

Derived the general form of smearing distributions preserving the semigroup property

Simplified Kramers-Moyal equations using superpositions of distributions

Extended superposition techniques to quantum mechanics

Abstract

Probability distributions which can be obtained from superpositions of Gaussian distributions of different variances v = \sigma ^2 play a favored role in quantum theory and financial markets. Such superpositions need not necessarily obey the Chapman-Kolmogorov semigroup relation for Markovian processes because they may introduce memory effects. We derive the general form of the smearing distributions in v which do not destroy the semigroup property. The smearing technique has two immediate applications. It permits simplifying the system of Kramers-Moyal equations for smeared and unsmeared conditional probabilities, and can be conveniently implemented in the path integral calculus. In many cases, the superposition of path integrals can be evaluated much easier than the initial path integral. Three simple examples are presented, and it is shown how the technique is extended to quantum mechanics.