A q-Generalization of Product Densities and Janossy Functions in Stochastic Point Processes

TL;DR

This paper introduces a q-generalization of product densities and Janossy functions in stochastic point processes, revealing inherent correlations and extending classical combinatorial coefficients.

Contribution

It develops a new q-generalized framework for product densities and Janossy functions, connecting them with q-Stirling numbers and q-Poisson distributions.

Findings

q-product densities exhibit inherent correlations

q-generalized Janossy functions are closely related to q-product densities

The framework extends classical stochastic point process theory

Abstract

A q-generalization of the product densities in stochastic point processes is developed. The properties of these functions are studied and a q-generalization of the usual $C^r_s$ coefficients is obtained. This for fixed q-number of particles coincides with the q-Stirling numbers of the second kind. The q-product densities are investigated using q-Poisson distribution and this shows that the stochastic point processes involving consistent q-generalization are inherently correlated. A closely related function to q-product densities is a q-generalized Janossy function and a relation between the two is established.