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

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 Poisson–Mandel transform. Open problems are addressed at the end of the paper

On multivariable cumulant polynomial sequences with applications

“…Then the i-th cumulant polynomial E[C i, N ,1 (I 1 + · · · + I n )] represents the i-th cumulant of a mixed Poisson distribution with random parameter I 1 + · · · + I n . When I is the diagonal of a Wishart random matrix then N , 1 is employed in photocounting [5] and gives the number of electrons ejected by n pixels hit by a certain number of light waves.…”

“…When I is the diagonal of a Wishart random matrix then N , 1 is employed in photocounting [5] and gives the number of electrons ejected by n pixels hit by a certain number of light waves.…”

kStatistics: Unbiased Estimates of Joint Cumulant Products from the Multivariate Faà Di Bruno's Formula

“…Among its various applications, we recall the cumulant polynomial sequences and their connection with special families of stochastic processes (E. Di Nardo 2016a). Indeed, cumulant polynomials allow us to compute moments and cumulants of multivariate Lévy processes (E. Di Nardo and Oliva 2011), subordinated multivariate Lévy processes (E. Di Nardo, Marena, and Semeraro 2020) and multivariate compound Poisson processes (E. Di Nardo 2016b). Further examples can be found in Reiner (1976), Shrivastava (2002), Withers and Nadarajah (2010) or Privault (2021).…”