We define the notion of conditionally monotone product as a part of conditionally free product, which naturally includes monotone and Boolean products. Then we define conditionally monotone cumulants which are useful to calculate the limit distributions in central limit theorem and Poisson’s law of small numbers. Moreover, we introduce deformed convolutions arising from the conditionally monoto...

Cumulants were first defined and studied by Danish scientist T. N. Thiele. He called them semi-invariants. The importance of cumulants comes from the observation that many properties of random variables can be better represented by cumulants than by moments. We refer to Brillinger [3] and Gnedenko and Kolmogorov [4] for further detailed probabilistic aspects on this topic. Given a random variab...

This paper 1 addresses performance issues in the source separation problem. By drawing on the theory of optimal statistic matching, we derive new contrast functions which are optimal among those involving a given set of cumulants. In low noise, the optimal combination of a particular set of cumulants are shown to be parameter independent and can be pre-computed. We give speciic exemples in clos...

Noise is a result of stochastic processes that originate from quantum or classical sources. Higher-order cumulants of the probability distribution underlying the stochastic events are believed to contain details that characterize the correlations within a given noise source and its interaction with the environment, but they are often difficult to measure. Here we report measurements of the tran...

in this paper, a new relationship linking cumulants and the coefficients of non linear systems is presented using non Gaussian signal input. This relationship is used to develop a new algorithm based on fourth order cumulants for the identification of the kernels of non linear systems in noiseless and noise environment case. The simulation results are presented to illustrate the performance of ...

High order cumulants carry information about statistics of non–normally distributed multivariate data. Such cumulants are utilised in extreme events analysis, small target detection or outliers detection. In this work we present a new algorithm,for updating high order cumulant tensors of random multivariate data, if new package of data is recorded. We show algebraically and numerically, that th...

The three-dimensional high-order simulation algorithm HOSIM is developed to simulate complex nonlinear and non-Gaussian systems. HOSIM is an alternative to the current MP approaches and it is based upon new high-order spatial connectivity measures, termed high-order spatial cumulants. The HOSIM algorithm implements a sequential simulation process, where local conditional distributions are gener...

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 ...