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Probability approximation by Clark-Ocone covariance representation. (English) Zbl 1308.60069

The authors first show how a general integration by parts formula combined with the Stein-Chen method allows to obtain bounds on the total variation distance or on the Wasserstein distance between a random variable and the Gaussian standard variable. With the same tools, there is an analogous result with some more intricate distance when the Gaussian variable is replaced by a centered gamma variable. On Wiener or Poisson spaces, instead of dealing with the usual covariance representation based on the Ornstein-Uhlenbeck or number operator and its inverse, they use the Clark-Ocone representation formula which provides a covariance identity which may be simpler to compute. Then, they focus on the Poisson space and succeed in approximating some Poisson jump times functionals and multiple Poisson stochastic integrals. Another application of the general method is given for the normal approximation in total variation of the compound Poisson distribution.

MSC:

60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60G57 Random measures
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05 Central limit and other weak theorems
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