Stein’s method on Wiener chaos

Density Formula and Concentration Inequalities with Malliavin Calculus

“…Our Section 4 can thus be thought as a way to extend the phenomena described in [11] when comparison with the normal distribution can only be expected to go one way. Theorem 4.3 shows that we may have no control over how heavy the tail of Z may be (beyond the existence of a second moment), but the condition g Z (Z) σ 2 > 0 P -a.s. essentially guarantees that it has to be no less heavy than a Gaussian tail with variance σ 2 .…”

“…It is useful to keep in mind throughout this paper that, by [11,Proposition 3.9], g Z (z) 0 for almost all z in the support of the law of Z.…”

“…By [11,Proposition 3.9], we know that g Z is non-negative on the support of the law of Z. Under some general conditions on Z (see Theorem 3.1 for a precise statement), the density ρ of the law of Z (provided it exists) is given by the following new formula, valid for almost all z in the support of ρ:…”

“…Remark 3.2 The integration by parts formula (3.15) was proved and used for the rst time by Nourdin and Peccati in [11], in order to perform error bounds in the normal approximation of Z. Specically, [11] shows, by combining Stein's method with (3.15), that, if Var(Z) > 0,…”

Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals

“…Malliavin-Stein method stands for the combination of two powerful tools in probability theory: Paul Malliavin's differential calculus and Charles Stein's method of normal approximation. This intersection originates from the seminal paper [12] by Nourdin and Peccati, who were able to associate a quantitative bound to the remarkable fourth moment theorem established by Nualart and Peccati [15] among many other things. For a comprehensive overview, one can refer to the website [11] and the recent monograph [13].…”