• Bernoulli
  • Volume 24, Number 4B (2018), 3384-3421.

The Gamma Stein equation and noncentral de Jong theorems

Christian Döbler and Giovanni Peccati

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We study the Stein equation associated with the one-dimensional Gamma distribution, and provide novel bounds, allowing one to effectively deal with test functions supported by the whole real line. We apply our estimates to derive new quantitative results involving random variables that are non-linear functionals of random fields, namely: (i) a non-central quantitative de Jong theorem for sequences of degenerate $U$-statistics satisfying minimal uniform integrability conditions, significantly extending previous findings by de Jong (J. Multivariate Anal. 34 (1990) 275–289), Nourdin, Peccati and Reinert (Ann. Probab. 38 (2010) 1947–1985) and Döbler and Peccati (Electron. J. Probab. 22 (2017) no. 2), (ii) a new Gamma approximation bound on the Poisson space, refining previous estimates by Peccati and Thäle (ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013) 525–560) and (iii) new Gamma bounds on a Gaussian space, strengthening estimates by Nourdin and Peccati (Probab. Theory Related Fields 145 (2009) 75–118). As a by-product of our analysis, we also deduce a new inequality for Gamma approximations via exchangeable pairs, that is of independent interest.

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Bernoulli, Volume 24, Number 4B (2018), 3384-3421.

Received: March 2017
Revised: June 2017
First available in Project Euclid: 18 April 2018

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de Jong theorem degenerate $U$-statistics exchangeable pairs Gamma approximation Hoeffding decomposition multiple stochastic integrals Stein equation Stein’s method


Döbler, Christian; Peccati, Giovanni. The Gamma Stein equation and noncentral de Jong theorems. Bernoulli 24 (2018), no. 4B, 3384--3421. doi:10.3150/17-BEJ963.

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