Nourdin-Peccati analysis on Wiener and Wiener-Poisson space for general distributions

Stein's method of exchangeable pairs for absolutely continuous, univariate distributions with applications to the Polya urn model

“…In [Sch01] the Stein equation is found, but no bounds on the solutions are given. Specialized to the Beta distributions, the general results from the papers [KT12] and [EV12] yield useful bounds on the Stein solutions and their first derivatives. Our bounds are qualitatively comparable to those from [EV12].…”

“…Thus, for concrete distributions and explicit constants it might therefore by useful to work with our bound from Corollary 2.16 (b). (iii) For the normal distribution and also for the larger class of distributions discussed in [EL10], one also has a bound of the form g ′′ h ∞ ≤ C h ′ ∞ for some finite constant C holding for each Lipschitz function h. As was shown by a universal example in [EV12], such a bound cannot be expected unless a…”

“…(ii) For concrete distributions the ratio appearing in the bound for g ′ h (x) may be bounded uniformly in x by some constant which can sometimes also be computed explicitely. Nevertheless, in [EV12] the authors give mild conditions for the existence of a finite constant k such that g ′ h ∞ ≤ k h ′ ∞ for any Lipschitzcontinuous h. In practice, these conditions are usually met. However, there is no hope of estimating the constant k by their method of proof.…”

Gamma limits and U-statistics on the Poisson space

“…The presence of the additional term (iv) Other relevant one-dimensional bounds for probabilistic approximations involving Malliavin operators on the Poisson space are proved in [22], dealing with normal approximations, [21], dealing with the Poisson approximation of integer-valued random variables and [7], focusing on absolutely continuous distributions whose support is given by the real line. See [2,24] for several multidimensional extensions.…”

“…[18]); see [2,21,34] for references based on the combination of Malliavin calculus and of the Chen-Stein method for Poisson approximations. We also refer to [7] for recent extensions to general absolutely continuous distributions having support equal to the real line.…”