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Wiener Chaos: Moments, Cumulants and Diagrams
A survey with Computer Implementation
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115 b
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This is a book about combinatorial structures related to the law of random variables belonging to a so-called Wiener chaos, that is, to a space of multiple stochastic integrals (of a given fixed order) associated with some infinitely divisible random measure. The combinatorial structures involved are those of lattices of partitions of finite sets, over which we define incidence algebras, Möbius functions and associated inversion formulae. As discussed in the text, this combinatorial standpoint (due to Rota and Wallstrom) provides an ideal framework in order to systematically deal with diagram formulae, that are graphical devices used to compute moments and cumulants of random variables. This systematic study of diagram formulae, moments and cumulants is unique in its genre, and can be only found in our text. Several application are described, in particular to recent limit theorems for chaotic random variables. An explicit computer implementation into MATHEMATICA completes the work
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