Noncentral convergence of multiple integrals

By Ivan Nourdin, Giovanni Peccati. 2007

84 citations recorded by the source corpus.

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.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ca0437ccd3c6110b908787b5f37c17d96d773271

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

In Mathematical Physics

Citations in the corpus

Multivariate normal approximation using Stein's method and Malliavin calculus (2008, methodology): “See [11] for an application of these techniques to non-central limit theorems.”
Cumulants on the Wiener Space (2009, methodology): “The starting point of this active line of research is the paper [21], where the authors use Malliavin calculus in order to refine some criteria for asymptotic normality on a fixed Wiener chaos, originally proved in [22, 24] (see also [ 12 ] for some non-central version of these results).”
Stein’s method on Wiener chaos (2007, methodology): “The same relation has been later applied in [33], where a characterization of non-central limit theorems for multiple integrals is provided.”
Stein’s method and exact Berry–Esseen asymptotics for functionals of Gaussian fields (2008, unknown): “See [19] and [20], Section 3.”
Second order Poincaré inequalities and CLTs on Wiener space (2008, background): “dings contained in [16, 20, 21, 23], and virtually closes a very fruitful circle of recent ideas linking Malliavin calculus, Stein’s method and central limit theorems (CLTs) on Wiener space (see also [15]). The role of contraction inequalities is further explored in Section 6, where we study CLTs fo”
Invariance Principles for Homogeneous Sums: Universality of Gaussian Wiener Chaos (2009, methodology): “8 (Chi-square limit theorem for chaotic sums, [11]).”
Chaos of a Markov operator and the fourth moment condition (2012, methodology): “Based upon this observation, recent work by Nourdin and Peccati [13, 14] develops the tool of the so-called Stein method (cf.”
Malliavin-Stein method for variance-gamma approximation on Wiener space (2014, unknown): “A first step in this direction is the paper [11] by Nourdin and Peccati, in which conditions on the sequence {Fn}n∈N have been derived, under which convergence towards a centred Gamma distribution takes place.”
Central Limit Theorems for Multiple Skorokhod Integrals (2010, background): “In another direction, Nourdin and Peccati [14] proved an extension of the above equivalence (i) – (iii) for a sequence of random variables {Fn, n ≥ 1} in a fixed qth Wiener chaos, q ≥ 2, where the limit law is 2Gν/2 − ν, Gν/2 being the Gamma distribution with parameter ν/2.”
Stein's method meets Malliavin calculus: a short survey with new estimates (2009, methodology): “The main findings of [16] are refined in [18] and [22], again by means of Stein’s method.”
Fourth Moment Theorems for Markov Diffusion Generators (2013, methodology): “We also mention [3] for extensions to the free probability setting, [15] for the multivariate setting, [7] for Gamma approximation, as well as [14] for the discrete setting.”
Asymptotic independence of multiple Wiener-It\^o integrals and the resulting limit laws (2011, unknown): “)− 2σ2 ij . Combining Steps 1-3 gives the desired conclusion. 4.3. A multivariate version of the convergence towards χ2. Here we will prove a multivariate extension of a result of Nourdin and Peccati [7]. Such an extension was an open problem as far as we know. In what follows, G(ν) will denote a ra”