Ivan Nourdin
University of Luxembourg
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Featured researches published by Ivan Nourdin.
Annales De L Institut Henri Poincare-probabilites Et Statistiques | 2010
Ivan Nourdin; Giovanni Peccati; Anthony Réveillac
We combine Steins method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main findings by Peccati and Tudor (2005), Nualart and Ortiz-Latorre (2007), Peccati (2007) and Nourdin and Peccati (2007b, 2008); in particular, they apply to approximations by means of Gaussian vectors with an arbitrary, positive definite covariance matrix. Among several examples, we provide an application to a functional version of the Breuer-Major CLT for fields subordinated to a fractional Brownian motion.
Archive | 2012
Ivan Nourdin
Fractional Brownian motion (fBm) is a stochastic process which deviates significantly from Brownian motion and semimartingales, and others classically used in probability theory. As a centered Gaussian process, it is characterized by the stationarity of its increments and a medium- or long-memory property which is in sharp contrast with martingales and Markov processes. FBm has become a popular choice for applications where classical processes cannot model these non-trivial properties; for instance long memory, which is also known as persistence, is of fundamental importance for financial data and in internet traffic. The mathematical theory of fBm is currently being developed vigorously by a number of stochastic analysts, in various directions, using complementary and sometimes competing tools. This book is concerned with several aspects of fBm, including the stochastic integration with respect to it, the study of its supremum and its appearance as limit of partial sums involving stationary sequences, to name but a few. The book is addressed to researchers and graduate students in probability and mathematical statistics. With very few exceptions (where precise references are given), every stated result is proved.
Annals of Probability | 2009
Ivan Nourdin; Giovanni Peccati
We show how to detect optimal Berry–Esseen bounds in the normal approximation of functionals of Gaussian fields. Our techniques are based on a combination of Malliavin calculus, Stein’s method and the method of moments and cumulants, and provide de facto local (one-term) Edgeworth expansions. The findings of the present paper represent a further refinement of the main results proven in Nourdin and Peccati [Probab. Theory Related Fields 145 (2009) 75–118]. Among several examples, we discuss three crucial applications: (i) to Toeplitz quadratic functionals of continuous-time stationary processes (extending results by Ginovyan [Probab. Theory Related Fields 100 (1994) 395–406] and Ginovyan and Sahakyan [Probab. Theory Related Fields 138 (2007) 551–579]); (ii) to “exploding” quadratic functionals of a Brownian sheet; and (iii) to a continuous-time version of the Breuer–Major CLT for functionals of a fractional Brownian motion.
Annals of Probability | 2009
Ivan Nourdin; Giovanni Peccati
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.
Annals of Probability | 2012
Todd Kemp; Ivan Nourdin; Giovanni Peccati; Roland Speicher
We prove that a normalized sequence of multiple Wigner integrals (in a fixed order of free Wigner chaos) converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2, the fourth moment of the semicircular law. This extends to the free probabilistic, setting some recent results by Nualart and Peccati on characterizations of central limit theorems in a fixed order of Gaussian Wiener chaos. Our proof is combinatorial, analyzing the relevant noncrossing partitions that control the moments of the integrals. We can also use these techniques to distinguish the first order of chaos from all others in terms of distributions; we then use tools from the free Malliavin calculus to give quantitative bounds on a distance between different orders of chaos. When applied to highly symmetric kernels, our results yield a new transfer principle, connecting central limit theorems in free Wigner chaos to those in Gaussian Wiener chaos. We use this to prove a new free version of an important classical theorem, the Breuer–Major theorem.
Annals of Probability | 2009
Ivan Nourdin; Anthony Réveillac
We derive the asymptotic behavior of weighted quadratic variations of fractional Brow- nian motion B with Hurst index H = 1/4. This completes the only missing case in a very recent work by I. Nourdin, D. Nualart and C.A. Tudor. Moreover, as an application, we solve a recent conjecture of K. Burdzy and J. Swanson on the asymptotic behavior of the Riemann sums with alternating signs associated to B.
Annals of Probability | 2008
Ivan Nourdin
This note is devoted to a fine study of the convergence of some weighted quadratic and cubic variations of a fractional Brownian motion B with Hurst index H in (0,1/2). With the help of Malliavin calculus, we show that, correctly renormalized, the weighted quadratic variation of B that we consider converges in L^2 to an explicit limit when H 1/4. In the same spirit, we also show that, correctly renormalized, the weighted cubic variation of B converges in L^2 to an explicit limit when H<1/6.
Electronic Journal of Statistics | 2009
Jean-Christophe Breton; Ivan Nourdin; Giovanni Peccati
In this short note, we show how to use concentration inequal- ities in order to build exact confidence intervals for the Hurst parameter associated with a one-dimensional fractional Brownian motion. AMS 2000 subject classifications: Primary 60G15; secondary 60F05,
arXiv: Probability | 2015
Ivan Nourdin; Giovanni Peccati
We compute the exact rates of convergence in total variation associated with the ‘fourth moment theorem’ by Nualart and Peccati (2005), stating that a sequence of random variables living in a fixed Wiener chaos verifies a central limit theorem (CLT) if and only if the sequence of the corresponding fourth cumulants converges to zero. We also provide an explicit illustration based on the Breuer-Major CLT for Gaussian-subordinated random sequences.
Annals of Probability | 2006
Ivan Nourdin; Thomas Simon
We consider the problem of absolute continuity for the one-dimensional SDE Xt=x+∫0ta(Xs) ds+Zt, where Z is a real Levy process without Brownian part and a a function of class