Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the fourth moment condition for Rademacher chaos

Christian Döbler and Kai Krokowski

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Abstract

Adapting the spectral viewpoint suggested in (Ann. Probab. 40 (6) (2012) 2439–2459) in the context of symmetric Markov diffusion generators and recently exploited in the non-diffusive setup of a Poisson random measure (Ann. Probab. (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 (Ann. Probab. 33 (1) (2005) 177–193) and Poisson (Ann. Probab. (2017)) situation, closely resembles that of degenerate, non-symmetric $U$-statistics from the classical paper (J. Multivariate Anal. 34 (2) (1990) 275–289).

Résumé

En adaptant le point de vue spectral proposé par Ledoux (Ann. Probab. 40 (6) (2012) 2439–2459) dans le cadre des générateurs des diffusions Markoviennes, qui a également été exploité récemment dans la situation non-diffusive d’une mesure aléatoire de Poisson (Ann. Probab. (2017)), nous étudions la condition du quatrième moment pour des intégrales multiples discrètes relatives à des suites de Rademacher générales, c.à.d. non-symétriques et non-homogènes, et nous démontrons que, dans ce cas, le quatrième moment ne gouverne pas complètement leur normalité asymptotique. En effet, il faut aussi tenir compte de l’influence maximale des fonctions de noyau correspondantes. En particulier, nous démontrons qu’il n’y a pas de théorème du quatrième moment exact pour des intégrales multiples discrètes de l’ordre $m\geq2$ relatives à une suite de Rademacher symétrique. Ce comportement, qui contraste avex les situations Gaussiennes (Ann. Probab. 33 (1) (2005) 177–193) et Poissoniennes (Ann. Probab. (2017)), ressemble fortement à celui des $U$-statistiques dégénerées et non-symétriques dans l’article classique (J. Multivariate Anal. 34 (2) (1990) 275–289).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 61-97.

Dates
Received: 6 June 2017
Revised: 15 November 2017
Accepted: 16 November 2017
First available in Project Euclid: 18 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1547802395

Digital Object Identifier
doi:10.1214/17-AIHP876

Zentralblatt MATH identifier
07039765

Subjects
Primary: 60F05: Central limit and other weak theorems 60H07: Stochastic calculus of variations and the Malliavin calculus 60H05: Stochastic integrals

Keywords
Fourth moment theorem Stein’s method Discrete Malliavin calculus Rademacher sequences Carré du champ operator

Citation

Döbler, Christian; Krokowski, Kai. On the fourth moment condition for Rademacher chaos. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 61--97. doi:10.1214/17-AIHP876. https://projecteuclid.org/euclid.aihp/1547802395


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