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A self-contained and probability-oriented introduction to the theory of lattice of partitions, with a unique software implementation that makes our book an ideal introduction to the field
A complete and self-contained combinatorial analysis of cumulants and diagram formulae, unique in its genre
An introduction to Wiener chaos, and a new combinatorial interpretation of recently proved limit theorems
The concept of Wiener chaos generalizes to an infinite-dimensional setting the
properties of orthogonal polynomials associated with probability distributions
on the real line. It plays a crucial role in modern probability theory, with applications
ranging from Malliavin calculus to stochastic differential equations and from
probabilistic approximations to mathematical finance.
This book is concerned with combinatorial structures arising from the study
of chaotic random variables related to infinitely divisible random measures.
The combinatorial structures involved are those of partitions of finite sets,
over which Möbius functions and related inversion formulae are defined.
This combinatorial standpoint (which is originally due to Rota and Wallstrom)
provides an ideal framework for diagrams, which are graphical devices used
to compute moments and cumulants of random variables.
Several applications are described, in particular, recent limit theorems for chaotic random variables.
An Appendix presents a computer implementation in MATHEMATICA for many of the formulae.
Content Level »Graduate
Keywords »Diagram formulae - Lattices of partitions - Limit theorems - Moments and cumulants - Wiener chaos