We provide an algebraic setting for cumulants and factorial moments through the classical umbral calculus. Main tools are the compositional inverse of the unity umbra, connected with the logarithmic power series, and a new umbra here introduced, the singleton umbra. Various formulae are given expressing cumulants, factorial moments and central moments by umbral functions.

A combinatorial formula is derived which expresses free cumulants in terms of classical comulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson distributions. The latter count connected pairings and connected partitions respectively. The proof relies on Möbius inversion on the partition lattice.

Generalizing the realized variance, skewness (Neuberger, 2012) and kurtosis (Bae Lee, 2020), we construct cumulants with so-called aggregation property. They are unbiased statistics of a martingale marginal based on sub-period increments its lower-order conditional cumulant processes. Our key finding is relation between property complete Bell polynomials. For an application give alternative pro...

Multivariate Gaussian data is completely characterized by its mean and covariance, yet modern non-Gaussian data makes higher-order statistics such as cumulants inevitable. For univariate data, the third and fourth scalar-valued cumulants are relatively well-studied as skewness and kurtosis. For multivariate data, these cumulants are tensor-valued, higher-order analogs of the covariance matrix c...

A formula expressing free cumulants in terms of the Jacobi parameters of the corresponding orthogonal polynomials is derived. It combines Flajolet's theory of continued fractions and Lagrange inversion. For the converse we discuss Gessel-Viennot theory to express Hankel determinants in terms of various cumulants.

We derive a formula which expresses a second order cumulant whose entries are products as a sum of cumulants where the entries are single factors. This extends to the second order case the formula of Krawczyk and Speicher. We apply our result to the problem of calculating the second order cumulants of a semicircular and Haar unitary operator.

In the context of random multiplicative cascade processes, we derive analytical solutions for oneand twopoint cumulants with restored translational invariance. On taking ratios of cumulants in ln ε, geometrical effects due to spatial averaging cancel out. These ratios can successfully distinguish between splitting functions while multifractal scaling exponents and multiplier distributions cannot.

The cumulants and factorial moments of photon distribution for squeezed and correlated light are calculated in terms of Chebyshev, Legendre and Laguerre polynomials. The phenomenon of strong oscillations of the ratio of the cumulant to factorial moment is found. Running title: Oscillations of cumulants in squeezed states.

We have measured secondand third-order cumulants in UA1 data (p̄p collisions at 630 GeV/c). Rather than quoting numerical values for source parameters, we have used these in three checks to test the “quantum statistics” theory for consistency over these cumulants. In the process, we have found a method for folding theoretical correlation functions with experimental one-particle distributions. Ou...