Asymptotic Cramer type decomposition for Wiener and Wigner integrals

We investigate generalizations of the Cramer theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.

Confidence intervals for the Hurst parameter of a fractional Brownian motion based on finite sample size

In this paper, we show how concentration inequalities for Gaussian quadratic form can be used to propose exact confidence intervals of the Hurst index parametrizing a fractional Brownian motion. Both cases where the scaling parameter of the fractional Brownian motion is known or unknown are investigated. These intervals are obtained by observing a single discretized sample path of a fractional Brownian motion and without any assumption on the parameter $H$.

Rescaled weighted random balls models and stable self-similar random fields

We consider weighted random balls in $real^d$ distributed according to a random Poisson measure with heavy-tailed intensity and study the asymptotic behaviour of the total weight of some configurations in $real^d$. This procedure amounts to be very rich and several regimes appear in the limit, depending on the intensity of the balls, the zooming factor, the tail parameters of the radii and of the weights. Statistical properties of the limit fields are also evidenced, such as isotropy, self-similarity or dependence. One regime is of particular interest and yields $alpha$-stable stationary isotropic self-similar generalized random fields which recovers Takenaka fields, Telecom process or fractional Brownian motion.

Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion

Let $qgeq 2$ be a positive integer, $B$ be a fractional Brownian motion with Hurst index $Hin(0,1)$, $Z$ be an Hermite random variable of index $q$, and $H_q$ denote the Hermite polynomial having degree $q$. For any $ngeq 1$, set $V_n=sum_{k=0}^{n-1} H_q(B_{k+1}-B_k)$. The aim of the current paper is to derive, in the case when the Hurst index verifies $H>1-1/(2q)$, an upper bound for the total variation distance between the laws $mathscr{L}(Z_n)$ and $mathscr{L}(Z)$, where $Z_n$ stands for the correct renormalization of $V_n$ which converges in distribution towards $Z$. Our results should be compared with those obtained recently by Nourdin and Peccati (2007) in the case when $H<1-1/(2q)$, corresponding to the situation where one has normal approximation.

Convex ordering for random vectors using predictable representation

We prove convex ordering results for random vectors admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component. Our method uses forward-backward stochastic calculus and extends previous results in the one-dimensional case. We also study a geometric interpretation of convex ordering for discrete measures in connection with the conditions set on the jump heights and intensities of the considered processes.