On the computation of classical, boolean and free cumulants
نویسندگان
چکیده
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 cumulants. This is realized by using a suitable polynomial representation of Abel polynomials. The algorithm relies on the classical umbral calculus, a symbolic language introduced by Rota and Taylor in [11], that is particularly suited to be implemented by using software for symbolic computations. Here we give a MAPLE procedure. Comparisons with existing procedures, especially for conversions between moments and free cumulants, as well as examples of applications to some well-known distributions (classical and free) end the paper. keywords: umbral calculus, classical cumulant, boolean cumulant, free cumulant, Abel polynomial AMS subject classification: 65C60, 05A40, 46L53
منابع مشابه
Relations between Cumulants in Noncommutative Probability
We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed.
متن کاملAppell Polynomials and Their Relatives Ii. Boolean Theory
The Appell-type polynomial family corresponding to the simplest non-commutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal non-commutative probability theories (the other two being free and tensor/classical probability). The basic properties of the Boolean Appell polynomials are described. In particular, their generatin...
متن کاملThe Monotone Cumulants
In the present paper we define the notion of generalized cumulants which gives a universal framework for commutative, free, Boolean, and especially, monotone probability theories. The uniqueness of generalized cumulants holds for each independence, and hence, generalized cumulants are equal to the usual cumulants in commutative, free and Boolean cases. The way we define (generalized) cumulants ...
متن کاملCumulants, free cumulants and half-shuffles.
Free cumulants were introduced as the proper analogue of classical cumulants in the theory of free probability. There is a mix of similarities and differences, when one considers the two families of cumulants. Whereas the combinatorics of classical cumulants is well expressed in terms of set partitions, that of free cumulants is described and often introduced in terms of non-crossing set partit...
متن کاملCumulants of a convolution and applications to monotone probability theory
In non-commutative probability theory, cumulants and their appropriate generating function are defined when a notion of independence is given [17, 18, 19]. They are useful especially in the central limit theorem and Poisson’s law of small numbers. We denote by any one of classical, free and boolean convolutions. We denote by Dλ the dilation operator defined by (Dλμ)(B) = μ(λ B) for any Borel se...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Applied Mathematics and Computation
دوره 208 شماره
صفحات -
تاریخ انتشار 2009