arXiv subjects

Corinne Berzin

Publications and source records attributed to Corinne Berzin.

3 recordsLinked to original sources

arXiv2022

Kac-Rice formula: A contemporary overview of the main results and applications

The book develops the fundamental ideas of the famous Kac-Rice formula for vectorvalued random fields. This formula allows to compute the expectation and moments of the measure, and integrals with respect to this measure, of the sets of levels of such fields. After a presentation of the historical context of the Kac-Rice formula, we give an elementary demonstration of the co-area formula. This formula replaces the change of variable formula in multiple integrals and a direct application of this formula gives the Kac-Rice formula for almost all levels. We emphasize the necessity of having the formula for all levels, because for some applications one needs, for example, the formula for level zero.

math.CA↗

arXiv2020

Estimation of Local Anisotropy Based on Level Sets

Consider an affine Gaussian field X : R 2 $\rightarrow$ R, that is a process equal in law to Z(At), where Z is isotropic and A : R2 $\rightarrow$ R2 is a self-adjoint definite positive matrix. Denote 0 < $λ$ = $λ$\_2 / $λ$\_1 \le 1 the ratio of the eigenvalues of A. This paper is aimed at testing the null hypothesis '' X is isotropic'' versus the alternative '' X is affine''. Roughly speaking, this amounts to testing '' $λ$ = 1 '' versus '' $λ$ < 1 ''. By setting level u in R, this is implemented by the partial observations of process X through some particular level functionals viewed over a square T, which grows to R2. This leads us to provide estimators for the affinity parameters that are shown to be almost surely consistent. Their asymptotic normality provide confidence intervals for parameters. This paper offered an important opportunity to study general level functionals near the level u, part of the difficulties arises from the fact that the topology of level set CT,X (u) = {t $\in$ T : X(t) = u} can be irregular, even if the trajectories of X are regular. A significant part of the paper is dedicated to show the L2-continuity in the level u of these general functionals.

math.PR↗

arXiv2008

Estimation in models driven by fractional Brownian motion

Let $\{b_H(t),t\in\mathbb{R}\}$ be the fractional Brownian motion with parameter $0<H<1$. When $1/2<H$, we consider diffusion equations of the type \[X(t)=c+\int_0^tσ\bigl(X(u)\bigr)\mathrm {d}b_H(u)+\int _0^tμ\bigl(X(u)\bigr)\mathrm {d}u.\] In different particular models where $σ(x)=σ$ or $σ(x)=σx$ and $μ(x)=μ$ or $μ(x)=μx$, we propose a central limit theorem for estimators of $H$ and of $σ$ based on regression methods. Then we give tests of the hypothesis on $σ$ for these models. We also consider functional estimation on $σ(\cdot)$ in the above more general models based in the asymptotic behavior of functionals of the 2nd-order increments of the fBm.

math.PR↗