On photon statistics parametrized by a non-central Wishart random matrix

TL;DR

This paper introduces polykays as unbiased estimators for photon statistics in photocounting, utilizing spectral statistics of non-central Wishart matrices and generalized Bell polynomials to improve parameter estimation and distribution approximation.

Contribution

It proposes a novel use of polykays for photon statistics estimation and links spectral properties of non-central Wishart matrices to photocounting distributions.

Findings

Polykays effectively estimate cumulants of photon counts.

Spectral statistics of Wishart matrices approximate photocounting distributions.

Multivariate polykays aid in Mendel-Poisson transform approximation.

Abstract

In order to tackle parameter estimation of photocounting distributions, polykays of acting intensities are proposed as a new tool for computing photon statistics. As unbiased estimators of cumulants, polykays are computationally feasible thanks to a symbolic method recently developed in dealing with sequences of moments. This method includes the so-called method of moments for random matrices and results to be particularly suited to deal with convolutions or random summations of random vectors. The overall photocounting effect on a deterministic number of pixels is introduced. A random number of pixels is also considered. The role played by spectral statistics of random matrices is highlighted in approximating the overall photocounting distribution when acting intensities are modeled by a non-central Wishart random matrix. Generalized complete Bell polynomials are used in order to compute joint moments and joint cumulants of multivariate photocounters. Multivariate polykays can be successfully employed in order to approximate the multivariate Mendel-Poisson transform. Open problems are addressed at the end of the paper.