Zakhar Kabluchko

Stationary max-stable fields associated to negative definite functions

Let Wi, i∈ℕ, be independent copies of a zero-mean Gaussian process {W(t), t∈ℝd} with stationary increments and variance σ2(t). Independently of Wi, let ∑i=1∞δUi be a Poisson point process on the real line with intensity e−y dy. We show that the law of the random family of functions {Vi(⋅), i∈ℕ}, where Vi(t)=Ui+Wi(t)−σ2(t)/2, is translation invariant. In particular, the process η(t)=⋁i=1∞Vi(t) is a stationary max-stable process with standard Gumbel margins. The process η arises as a limit of a suitably normalized and rescaled pointwise maximum of n i.i.d. stationary Gaussian processes as n→∞ if and only if W is a (nonisotropic) fractional Brownian motion on ℝd. Under suitable conditions on W, the process η has a mixed moving maxima representation.

Asymptotic distribution of complex zeros of random analytic functions

Let

Locally adaptive image denoising by a statistical multiresolution criterion

\xi_0,\xi_1,\ldots

Random marked sets

be independent identically distributed complex- valued random variables such that

Stationary systems of Gaussian processes

\mathbb{E}\log(1+|\xi _0|)<\infty

Roots of random polynomials whose coefficients have logarithmic tails

. We consider random analytic functions of the form \[\mathbf{G}_n(z)=\sum_{k=0}^{\infty}\xi_kf_{k,n}z^k,\] where

Convex hulls of random walks, hyperplane arrangements, and Weyl chambers

f_{k,n}

Distribution of levels in high-dimensional random landscapes

are deterministic complex coefficients. Let

Local universality for real roots of random trigonometric polynomials

\mu_n

A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk

be the random measure counting the complex zeros of