Deconvolution of point processes

TL;DR

This paper introduces a mathematical framework for deconvolving point processes by dividing their probability generating functionals, utilizing higher-order Gateaux differentials to recover individual processes from superpositions.

Contribution

It develops a higher-order quotient rule for Gateaux differentials and applies it to deconvolve superposed point processes, advancing the mathematical tools for point process analysis.

Findings

Derived a higher-order quotient rule for Gateaux differentials.

Applied the quotient rule to deconvolve superposed point processes.

Provided a theoretical foundation for point process deconvolution.

Abstract

The superposition of two independent point processes can be described by multiplication of their probability generating functionals (p.g.fl.s). The inverse operation, which can be viewed as a deconvolution, is defined by dividing the superposed process by one of its constituent p.g.fl.s. The deconvolved process is computed using the higher-order chain rule for Gateaux differentials. The higher-order quotient rule for Gateaux differentials is first established and then applied to point processes.