On the fourth moment condition for Rademacher chaos

Döbler, Christian; Krokowski, Kai

doi:10.48550/arxiv.1706.00751

Cited by 1 publication

(2 citation statements)

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“…However, there are examples of non-diffusive situations where no fourth moment theorem holds. Indeed, in the paper [11] the authors give counterexamples for the setup of infinite Rademacher sequences that show that fourth moment theorems do not always hold in this case but that one also has to take into account the maximal influence of every single member of the random sequence. Thus, we emphasize that we are here dealing with a very peculiar example of a non-diffusive Markov generator where, quite coincidentally, all relevant terms can be reduced to just the fourth cumulant.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The techniques developed in [26] have also been adapted to non-Gaussian spaces which admit a Malliavin calculus structure: for instance, the papers [16,34,36,41] deal with the Poisson space case, whereas [17,18,30,44] develop the corresponding techniques for sequences of independent Rademacher random variables. The question about general fourth moment theorems on these spaces, however, has remained open in general, until the two recent articles [13] and [11].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Fourth moment theorems on the Poisson space in any dimension

Döbler¹,

Vidotto²,

Zheng³

2018

Electron. J. Probab.

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We extend to any dimension the quantitative fourth moment theorem on the Poisson setting, recently proved by C. . In particular, by adapting the exchangeable pairs couplings construction introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove our results under the weakest possible assumption of finite fourth moments. This yields a Peccati-Tudor type theorem, as well as an optimal improvement in the univariate case.Finally, a transfer principle "from-Poisson-to-Gaussian" is derived, which is closely related to the universality phenomenon for homogeneous multilinear sums.where d TV denotes the total variation distance between the laws of two real random variables. The techniques developed in [26] have also been adapted to non-Gaussian spaces which admit a Malliavin calculus structure: for instance, the papers [16,34,36,41] deal with the Poisson space case, whereas [17,18,30,44] develop the corresponding techniques for sequences of independent Rademacher random variables. The question FOURTH MOMENT THEOREMS 3 about general fourth moment theorems on these spaces, however, has remained open in general, until the two recent articles [13] and [11].

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Section: Introduction and Main Resultsmentioning

confidence: 99%