Noncentral convergence of multiple integrals

@article{Nourdin2009NoncentralCO,
  title={Noncentral convergence of multiple integrals},
  author={I. Nourdin and G. Peccati},
  journal={Annals of Probability},
  year={2009},
  volume={37},
  pages={1412-1426}
}
Fix ν>0, denote by G(v/2) a Gamma random variable with parameter v/2, and let n≥2 be a fixed even integer. Consider a sequence (F_k) of square integrable random variables, belonging to the nth Wiener chaos of a given Gaussian process and with variance converging to 2v. As k goes to infinity, we prove that F_k converges in distribution to 2G(v/2)-v if and only if E(F_k^4)-12 E(F_k^3) tends to 12v^2-48v. 
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