D. Senato

Umbral nature of the Poisson random variables

The symbolic method, nowadays known as umbral calculus, has been extensively used since the nineteenth century although the mathematical community was sceptical of it, perhaps because of its lack of foundation. This method was fully developed by Rev. John Blissard in a series of papers beginning in 1861 [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]; nevertheless it is impossible to attribute the credit for the original idea to him alone since Blissard’s calculus has a mathematical source in symbolic differentiation. In [22] Lucas even claimed that the umbral calculus has its historical roots in the writing of Leibniz for the successive derivatives of a product with two or more factors; moreover Lucas held that this symbolic method had been developed subsequently by Laplace, by Vandermonde, by Herschel, and augmented by the works of Cayley and of Sylvester in the theory of forms. Lucas’ papers attracted considerable attention and the predominant contribution of Blissard to this method was kept in the background. Bell reviewed the whole subject in several papers, restoring the purport of the Blissard’s idea [4], and in 1940 he tried to give a rigorous foundation for the mystery at the bottom of the umbral calculus [5] but his attempt did not gain a hold. Indeed, in the first modern textbook of combinatorics [24], Riordan largely employed this symbolic method without giving any formal justification. It was Gian-Carlo Rota [26] who six years later disclosed the “umbral magic art” consisting in lowering and raising exponents bringing to the light the underlying linear functional. The ideas from [26] led Rota and his collaborators to conceive a beautiful theory which gave rise to a large variety of applications ([23, 27]). Some years later, Roman and Rota [25] gave rigorous form to the umbral tricks in the setting of Hopf algebras. On the other hand, as Rota himself has written in [28]: “…Although the notation of Hopf algebra satisfied the most ardent advocate of spic-and-span rigor, the translation of “classical”, umbral calculus into the newly found rigorous language made the method altogether unwieldy and unmanageable. Not only was the eerie feeling of witchcraft lost in the translation, but, after such a translation, the use of calculus to simplify computation and sharpen our intuition was lost by the wayside…” Thus in 1994 Rota and Taylor [28] started a rigorous and simple presentation of the umbral calculus in the spirit of the founders. The present article takes this last point of view.

An umbral setting for cumulants and factorial moments

We provide an algebraic setting for cumulants and factorial moments via the classical umbral calculus. Our main tools are the compositional inverse of the unity umbra, this being related to logarithmic power series, and a new umbra here introduced, the singleton umbra. We develop formulae that express cumulants, factorial moments and central moments as umbral functions.

A unifying framework for k-statistics, polykays and their multivariate generalizations

Through the classical umbral calculus, we provide a unifying syntax for single and multivariate kstatistics, polykays and multivariate polykays. From a combinatorial point of view, we revisit the theory as exposed by Stuart and Ord, taking into account the Doubilet approach to symmetric functions. Moreover, by using exponential polynomials rather than set partitions, we provide a new formula for k-statistics that results in a very fast algorithm to generate such estimators.

Cumulants and convolutions via Abel polynomials

We provide a unifying polynomial expression giving moments in terms of cumulants, and vice versa, holding in the classical, boolean and free setting. This is done by using a symbolic treatment of Abel polynomials. As a by-product, we show that in the free cumulant theory the volume polynomial of Pitman and Stanley plays the role of the complete Bell exponential polynomial in the classical theory. Moreover, via generalized Abel polynomials we construct a new class of cumulants, including the classical, boolean and free ones, and the convolutions linearized by them. Finally, via an umbral Fourier transform, we state an explicit connection between boolean and free convolution.

A new method for fast computing unbiased estimators of cumulants

We propose new algorithms for generating k-statistics, multivariate k-statistics, polykays and multivariate polykays. The resulting computational times are very fast compared with procedures existing in the literature. Such speeding up is obtained by means of a symbolic method arising from the classical umbral calculus. The classical umbral calculus is a light syntax that involves only elementary rules to managing sequences of numbers or polynomials. The cornerstone of the procedures here introduced is the connection between cumulants of a random variable and a suitable compound Poisson random variable. Such a connection holds also for multivariate random variables.

A new algorithm for computing the multivariate Faà di Bruno’s formula

Abstract A new algorithm for computing the multivariate Faa di Bruno’s formula is provided. We use a symbolic approach based on the classical umbral calculus that turns the computation of the multivariate Faa di Bruno’s formula into a suitable multinomial expansion. We propose a MAPLE procedure whose computational times are faster compared with the ones existing in the literature. Some illustrative applications are also provided.

Symbolic computation of moments of sampling distributions

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 procedures. Comparisons of computational times with known procedures show how this approach turns out to be more efficient in eliminating much unnecessary computation.

Natural statistics for spectral samples

Spectral sampling is associated with the group of unitary transformations acting on matrices in much the same way that simple random sampling is associated with the symmetric group acting on vectors. This parallel extends to symmetric functions, k-statistics and polykays. We construct spectral k-statistics as unbiased estimators of cumulants of trace powers of a suitable random matrix. Moreover we define normalized spectral polykays in such a way that when the sampling is from an infinite population they return products of free cumulants.

A symbolic method for k-statistics

Through the classical umbral calculus, we provide new, compact and easy to handle expressions for k-statistics, and more generally for U-statistics. In addition, this symbolic method can be naturally extended to multivariate cases and to generalized k-statistics.

Symbolic solutions of some linear recurrences

Abstract A symbolic method for solving linear recurrences of combinatorial and statistical interest is introduced. This method essentially relies on a representation of polynomial sequences as moments of a symbol that looks as the framework of a random variable with no reference to any probability space. We give several examples of applications and state an explicit form for the class of linear recurrences involving Sheffer sequences satisfying a special initial condition. The results here presented can be easily implemented in a symbolic software.