On the fourth moment condition for Rademacher chaos
Autor: | Kai Krokowski, Christian Döbler |
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Jazyk: | angličtina |
Rok vydání: | 2019 |
Předmět: |
Statistics and Probability
Statistics::Theory Probability (math.PR) Stein’s method 0102 computer and information sciences Rademacher sequences 01 natural sciences 010104 statistics & probability 60H07 Fourth moment Mathematics::Probability Carré du champ operator 010201 computation theory & mathematics 60H05 60F05 FOS: Mathematics Discrete Malliavin calculus Fourth moment theorem 0101 mathematics Statistics Probability and Uncertainty Humanities Mathematics - Probability Mathematics |
Zdroj: | Ann. Inst. H. Poincaré Probab. Statist. 55, no. 1 (2019), 61-97 |
Popis: | Adapting the spectral viewpoint suggested in Ledoux (2012) in the context of symmetric Markov diffusion generators and recently exploited in the non-diffusive setup of a Poisson random measure by D\"obler and Peccati (2017), we investigate the fourth moment condition for discrete multiple integrals with respect to general, i.e.\ non-symmetric and non-homogeneous, Rademacher sequences and show that, in this situation, the fourth moment alone does not govern the asymptotic normality. Indeed, here one also has to take into consideration the maximal influence of the corresponding kernel functions. In particular, we show that there is no exact fourth moment theorem for discrete multiple integrals of order $m\geq2$ with respect to a symmetric Rademacher sequence. This behavior, which is in contrast to the Gaussian (see Nualart and Peccati (2005)) and Poisson (see D\"obler and Peccati (2017)) situation, closely resembles the conditions for asymptotic normality of degenerate, non-symmetric $U$-statistics from the classical paper by de Jong (1990). Comment: 40 pages; complete characterization of the symmetric case added; alternative proof extended to the general case; some mistakes fixed |
Databáze: | OpenAIRE |
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