Moments, cumulants and diagram formulae for non-linear functionals of random measures

Cumulants on the Wiener Space

We combine infinite-dimensional integration by parts procedures with a recursive relation on moments (reminiscent of a formula by Barbour (1986)), and deduce explicit expressions for cumulants of functionals of a general Gaussian field. These findings yield a compact formula for cumulants on a fixed Wiener chaos, virtually replacing the usual “graph/diagram computations” adopted in most of the ...

A Representation for Characteristic Functionals of Stable Random Measures with Values in Sazonov Spaces

An umbral setting for cumulants and factorial moments

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.

Cumulants in Noncommutative Probability Theory I. Noncommutative Exchangeability Systems

Cumulants linearize convolution of measures. We use a formula of Good to define noncommutative cumulants in a very general setting. It turns out that the essential property needed is exchangeability of random variables. Roughly speaking the formula says that cumulants are moments of a certain “discrete Fourier transform” of a random variable. This provides a simple unified method to understand ...

Noisy Random Resistor Networks: Renormalized Field Theory for Multifractals in Percolation

We study the multifractal moments of the current distribution in a randomly diluted resistor networks with microscopic noise. These moments are related to the noise cumulants C (l) R (x, x ′) of the resistance between two sites x and x′ located on the same cluster by Cohn’s theorem. Our renormalized field theory is based on aD×E-fold replicated Hamiltonian introduced by Park, Harris and Lubensk...