We investigate the problem of finding necessary and sufficient conditions for convergence in distribution towards a general finite linear combination of independent chi-squared random variables, within the framework of random objects living on a fixed Gaussian space. Using a recent representation of cumulants in terms of the Malliavin calculus operators Γ i (introduced by Nourdin and Peccati in [13]), we provide conditions that apply to random variables living in a finite sum of Wiener chaoses. As an important by-product of our analysis, we shall derive a new proof and a new interpretation of a recent finding by Nourdin and Poly [16], concerning the limiting behaviour of random variables living in a Wiener chaos of order two. Our analysis contributes to a fertile line of research, that originates from questions raised by Marc Yor, in the framework of limit theorems for non-linear functionals of Brownian local times.

A. Inspired by the insightful article [4], we revisit the Nualart-Peccati-criterion [13] (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion generators. We are not only able to drastically simplify all of its previous proofs, but also to provide new settings of diffusive generators (Laguerre, Jacobi) where such a criterion holds. Convergence towards gamma and beta distributions under moment conditions is also discussed.

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Abstract. The continuity of Gaussian processes is extensively studied topic and it culminates in the Talagrand's notion of majorizing measures that gives complicated necessary and sufficient conditions. In this note we study the Hölder continuity of Gaussian processes. It turns out that necessary and sufficient conditions can be stated in a simple form that is a variant of the celebrated Kolmogorov-Čentsov condition.

Abstract. Fractional Ornstein-Uhlenbeck process of the second kind (fOU2) is solution of the Langevin equation dXt = −θXt dt+dY , we prove that the least squares estimator θT introduced in [[7], Statist. Probab. Lett. 80, no. 11-12, 1030Lett. 80, no. 11-12, -1038, provides a consistent estimator. Moreover, using central limit theorem for multiple Wiener integrals, we prove asymptotic normality of the estimator valid for the whole range H ∈ ( 1 2 , 1).

The celebrated Nualart-Peccati criterion [Ann. Probab. 33 (2005) 177-193] ensures the convergence in distribution toward a standard Gaussian random variable N of a given sequence {Xn} n≥1 of multiple Wiener-Itô integrals of fixed order, if E[X

Fractional Ornstein-Uhlenbeck process of the second kind (fOU 2 ) is solution of the Langevin equation dX t = −θX t dt + dY . Then using the ergodicity of fOU 2 process, we construct consistent estimators of drift parameter θ based on discrete observations in two possible cases: (i) the Hurst parameter H is known and (ii) the Hurst parameter H is unknown. Moreover, using Malliavin calculus technique, we prove central limit theorems for our estimators which is valid for the whole range H ∈ ( 1 2 , 1).

We provide a bound on a natural distance between finitely and infinitely supported elements of the unit sphere of ℓ 2 (IN ⋆ ), the space of real valued sequences with finite ℓ 2 norm. We use this bound to estimate the 2-Wasserstein distance between random variables which can be represented as linear combinations of independent random variables. Our results are expressed in terms of a discrepancy measure which is related to Nourdin and Peccati's Malliavin-Stein method. The main area of application of our results is towards the computation of quantitative rates of convergence towards elements of the second Wiener chaos. After particularizing our bounds to this setting and comparing them with the available literature on the subject (particularly the Malliavin-Stein method for variance-gamma random variables), we illustrate their versatility by tackling three examples: chi-squared approximation for second order U -statistics, asymptotics for sequences of quadratic forms and the behavior of the generalized Rosenblatt process at extreme critical exponent.