• Giovanni Peccati
  • Murad S. Taqqu
Part of the Bocconi & Springer Series book series (BS, volume 1)


The aim of this work is to provide a unified treatment of moments and cumulants associated with non-linear functionals of completely random measures. A “completely random measure” (also called an “independently scattered random measure”) is a measure ϕ with values in a space of random variables, such that ϕ(A) and ϕ(B) are independent random variables, whenever A and B are disjoint sets. Examples are Gaussian, Poisson or Gamma random measures. We will specifically focus on multiple stochastic integrals with respect to the random measure ϕ. These integrals are of the form
$$ \int\limits_\sigma {f(z_1 ,...,z_n )\phi (dz_1 )...\phi (dz_n )} $$
$$ \int\limits_{ \ge \sigma } {f(z_1 ,...,z_n )\phi (dz_1 )...\phi (dz_n )} , $$
where f is a symmetric function and ϕ is a completely random measure (for instance, Poisson or Gaussian) on the real line. The integration is not over all of ℝ n , but over a “diagonal” subset of ℝ n defined by a partition σ of the integers, …,n as illustrated below.


Random Measure Stochastic Integration Free Probability Malliavin Calculus Poisson Measure 
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Copyright information

© Springer-Verlag Italia 2011

Authors and Affiliations

  • Giovanni Peccati
    • 1
  • Murad S. Taqqu
    • 2
  1. 1.Mathematics Research UnitUniversity of LuxembourgLuxembourg
  2. 2.Department of Mathematics and StatisticsBoston UniversityBoston

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