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Extending the rigorous presentation of the " classical umbral calculus " [28], the so-called partition polynomials are interpreted with the aim to point out the umbral nature of the Poisson random variables. Among the new umbrae introduced , the main tool is the partition umbra that leads also to a simple expression of the functional composition of the… (More)

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… (More)

Motivated by a typical and well-known problem of neurobio-logical modeling, a parallel algorithm devised to simulate sample paths of stationary normal processes with rational spectral densities is implemented to evaluate first passage time probability densities for time-varying boundaries. After a self-contained outline of the original problem and of the… (More)

By means of the notion of umbrae indexed by multisets, a general method to express estimators and their products in terms of power sums is derived. A connection between the notion of multiset and integer partition leads immediately to a way to speed up the procedures. Comparisons of computational times with known procedures show how this approach turns out… (More)

Making use of a Rice-like series expansion, for a class of stationary Gaussian processes the asymp-totic behavior of the first passage time probability density function through certain time-varying boundaries, including periodic boundaries, is determined. Sufficient conditions are then given such that the density asymptotically exhibits an exponential… (More)

This paper introduces a simple and computationally efficient algorithm for conversion formulae between moments and cumulants. The algorithm provides just one formula for classical, boolean and free cu-mulants. This is realized by using a suitable polynomial representation of Abel polynomials. The algorithm relies on the classical umbral calculus , a… (More)