arXiv subjects

Zakhar Kabluchko

Publications and source records attributed to Zakhar Kabluchko.

At least 163 records · Page 9Linked to original sources

arXiv2012

Complex Random Energy Model: Zeros and Fluctuations

The partition function of the random energy model at inverse temperature $β$ is a sum of random exponentials $Z_N(β)=\sum_{k=1}^N \exp(β\sqrt{n} X_k)$, where $X_1,X_2,...$ are independent real standard normal random variables (= random energies), and $n=\log N$. We study the large $N$ limit of the partition function viewed as an analytic function of the complex variable $β$. We identify the asymptotic structure of complex zeros of the partition function confirming and extending predictions made in the theoretical physics literature. We prove limit theorems for the random partition function at complex $β$, both on the logarithmic scale and on the level of limiting distributions. Our results cover also the case of the sums of independent identically distributed random exponentials with any given correlations between the real and imaginary parts of the random exponent.

math.PR↗

arXiv2012

Critical points of random polynomials with independent identically distributed roots

Let $X_1,X_2,...$ be independent identically distributed random variables with values in $\C$. Denote by $μ$ the probability distribution of $X_1$. Consider a random polynomial $P_n(z)=(z-X_1)...(z-X_n)$. We prove a conjecture of Pemantle and Rivin [arXiv:1109.5975] that the empirical measure $μ_n:=\frac 1{n-1}\sum_{P_n'(z)=0} δ_z$ counting the complex zeros of the derivative $P_n'$ converges in probability to $μ$, as $n\to\infty$.

math.PR↗

arXiv2012

Estimation of Huesler-Reiss distributions and Brown-Resnick processes

Estimation of extreme-value parameters from observations in the max-domain of attraction (MDA) of a multivariate max-stable distribution commonly uses aggregated data such as block maxima. Since we expect that additional information is contained in the non-aggregated, single "large" observations, we introduce a new approach of inference based on a multivariate peaks-over-threshold method. We show that for any process in the MDA of the frequently used Hüsler-Reiss model or its spatial extension, the Brown-Resnick process, suitably defined conditional increments asymptotically follow a multivariate Gaussian distribution. This leads to computationally efficient estimates of the Hüsler-Reiss parameter matrix. Further, the results enable parametric inference for Brown-Resnick processes. A simulation study compares the performance of the new estimators to other commonly used methods. As an application, we fit a non-isotropic Brown-Resnick process to the extremes of 12 year data of daily wind speed measurements.

stat.ME↗

arXiv2012

Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian random fields

Consider a $d\times d$ matrix $M$ whose rows are independent centered non-degenerate Gaussian vectors $ξ_1,...,ξ_d$ with covariance matrices $Σ_1,...,Σ_d$. Denote by $\mathcal{E}_i$ the location-dispersion ellipsoid of $ξ_i:\mathcal{E}_i={\mathbf{x}\in\mathbb{R}^d : \mathbf{x}^\topΣ_i^{-1} \mathbf{x}\leqslant1}$. We show that $$ \mathbb{E}\,|\det M|=\frac{d!}{(2π)^{d/2}}V_d(\mathcal{E}_1,...,\mathcal{E}_d), $$ where $V_d(\cdot,...,\cdot)$ denotes the {\it mixed volume}. We also generalize this result to the case of rectangular matrices. As a direct corollary we get an analytic expression for the mixed volume of $d$ arbitrary ellipsoids in $\mathbb{R}^d$. As another application, we consider a smooth centered non-degenerate Gaussian random field $X=(X_1,...,X_k)^\top:\mathbb{R}^d\to\mathbb{R}^k$. Using Kac-Rice formula, we obtain the geometric interpretation of the intensity of zeros of $X$ in terms of the mixed volume of location-dispersion ellipsoids of the gradients of $X_i/\sqrt{\mathbf{Var} X_i}$. This relates zero sets of equations to mixed volumes in a way which is reminiscent of the well-known Bernstein theorem about the number of solutions of the typical system of algebraic equations.

math.PR↗

arXiv2012

Universality for zeros of random analytic functions

Let $ξ_0,ξ_1,...$ be independent identically distributed (i.i.d.) random variables such that $\E \log (1+|ξ_0|)<\infty$. We consider random analytic functions of the form $$ G_n(z)=\sum_{k=0}^{\infty} ξ_k f_{k,n} z^k, $$ where $f_{k,n}$ are deterministic complex coefficients. Let $ν_n$ be the random measure assigning the same weight $1/n$ to each complex zero of $G_n$. Assuming essentially that $-\frac 1n \log f_{[tn], n}\to u(t)$ as $n\to\infty$, where $u(t)$ is some function, we show that the measure $ν_n$ converges weakly to some deterministic measure which is characterized in terms of the Legendre--Fenchel transform of $u$. The limiting measure is universal, that is it does not depend on the distribution of the $ξ_k$'s. This result is applied to several ensembles of random analytic functions including the ensembles corresponding to the three two-dimensional geometries of constant curvature. As another application, we prove a random polynomial analogue of the circular law for random matrices.

math.PR↗

arXiv2012

Distribution of levels in high-dimensional random landscapes

We prove empirical central limit theorems for the distribution of levels of various random fields defined on high-dimensional discrete structures as the dimension of the structure goes to $\infty$. The random fields considered include costs of assignments, weights of Hamiltonian cycles and spanning trees, energies of directed polymers, locations of particles in the branching random walk, as well as energies in the Sherrington--Kirkpatrick and Edwards--Anderson models. The distribution of levels in all models listed above is shown to be essentially the same as in a stationary Gaussian process with regularly varying nonsummable covariance function. This type of behavior is different from the Brownian bridge-type limit known for independent or stationary weakly dependent sequences of random variables.

math.PR↗

arXiv2012

Random Marked Sets

We aim to link random fields and marked point processes and therefore introduce a new class of stochastic processes which are defined on a random set in R^d. Unlike for random fields, the mark covariance function of a marked random set is in general not positive definite. This implies that in many situations the use of simple geostatistical methods appears to be questionable. Surprisingly, for a special class of processes based on Gaussian random fields, we do have positive definiteness for the corresponding mark covariance function and mark correlation function.

math.PR↗

arXiv2011

Functional limit theorems for sums of independent geometric Lévy processes

Let $ξ_i$, $i\in \mathbb {N}$, be independent copies of a Lévy process $\{ξ(t),t\geq0\}$. Motivated by the results obtained previously in the context of the random energy model, we prove functional limit theorems for the process \[Z_N(t)=\sum_{i=1}^N\mathrm{e}^{ξ_i(s_N+t)}\] as $N\to\infty$, where $s_N$ is a non-negative sequence converging to $+\infty$. The limiting process depends heavily on the growth rate of the sequence $s_N$. If $s_N$ grows slowly in the sense that $\liminf_{N\to\infty}\log N/s_N>λ_2$ for some critical value $λ_2>0$, then the limit is an Ornstein--Uhlenbeck process. However, if $λ:=\lim_{N\to\infty}\log N/s_N\in(0,λ_2)$, then the limit is a certain completely asymmetric $α$-stable process $\mathbb {Y}_{α;ξ}$.

math.PR↗

arXiv2011

Persistence of competing systems of branching random walks

We consider a system of independent branching random walks on $\R$ which start off a Poisson point process with intensity of the form $e_λ(du)=e^{-λu}du$, where $λ\in\R$ is chosen in such a way that the overall intensity of particles is preserved. Denote by $χ$ the cluster distribution and let $ϕ$ be the log-Laplace transform of the intensity of $χ$. If $λϕ'(λ)>0$, we show that the system is persistent (stable) meaning that the point process formed by the particles in the $n$-th generation converges as $n\to\infty$ to a non-trivial point process $Π_{e_λ}^χ$ with intensity $e_λ$. If $λϕ'(λ)<0$, then the branching population suffers local extinction meaning that the limiting point process is empty. We characterize (generally, non-stationary) point processes on $\R$ which are cluster-invariant with respect to the cluster distribution $χ$ as mixtures of the point processes $Π_{ce_λ}^χ$ over $c>0$ and $λ\in K_{\text{st}}$, where $K_{\text{st}}=\{λ\in\R: ϕ(λ)=0, λϕ'(λ)>0\}$.

math.PR↗

arXiv2010

Stationary systems of Gaussian processes

We describe all countable particle systems on $\mathbb{R}$ which have the following three properties: independence, Gaussianity and stationarity. More precisely, we consider particles on the real line starting at the points of a Poisson point process with intensity measure $\mathfrak{m}$ and moving independently of each other according to the law of some Gaussian process $ξ$. We classify all pairs $(\mathfrak{m},ξ)$ generating a stationary particle system, obtaining three families of examples. In the first, trivial family, the measure $\mathfrak{m}$ is arbitrary, whereas the process $ξ$ is stationary. In the second family, the measure $\mathfrak{m}$ is a multiple of the Lebesgue measure, and $ξ$ is essentially a Gaussian stationary increment process with linear drift. In the third, most interesting family, the measure $\mathfrak{m}$ has a density of the form $αe^{-λx}$, where $α>0$, $λ\in\mathbb{R}$, whereas the process $ξ$ is of the form $ξ(t)=W(t)-λσ^2(t)/2+c$, where $W$ is a zero-mean Gaussian process with stationary increments, $σ^2(t)=\operatorname {Var}W(t)$, and $c\in\mathbb{R}$.

math.PR↗

arXiv2010

The maximum of the Gaussian $1/f^α$-noise in the case $α<1$

We prove that the appropriately normalized maximum of the Gaussian $1/f^α$-noise with $α<1$ converges in distribution to the Gumbel double-exponential law.

math.PR↗

arXiv2010

Extremes of the standardized Gaussian noise

Let $\{ξ_n, n\in\Z^d\}$ be a $d$-dimensional array of i.i.d. Gaussian random variables and define $\SSS(A)=\sum_{n\in A} ξ_n$, where $A$ is a finite subset of $\Z^d$. We prove that the appropriately normalized maximum of $\SSS(A)/\sqrt{|A|}$, where $A$ ranges over all discrete cubes or rectangles contained in $\{1,\ldots,n\}^d$, converges in the weak sense to the Gumbel extreme-value distribution as $n\to\infty$. We also prove continuous-time counterparts of these results.

math.PR↗

arXiv2010

Limit laws for sums of independent random products: the lattice case

Let $\{V_{i,j}; (i,j)\in\N^2\}$ be a two-dimensional array of i.i.d.\ random variables. The limit laws of the sum of independent random products $$ Z_n=\sum_{i=1}^{N_n} \prod_{j=1}^{n} e^{V_{i,j}} $$ as $n,N_n\to\infty$ have been investigated by a number of authors. Depending on the growth rate of $N_n$, the random variable $Z_n$ obeys a central limit theorem, or has limiting $α$-stable distribution. The latter result is true for non-lattice $V_{i,j}$ only. Our aim is to study the lattice case. We prove that although the (suitably normalized) sequence $Z_n$ fails to converge in distribution, it is relatively compact in the weak topology, and describe its cluster set. This set is a topological circle consisting of semi-stable distributions.

math.PR↗

arXiv2010

Locally adaptive image denoising by a statistical multiresolution criterion

We demonstrate how one can choose the smoothing parameter in image denoising by a statistical multiresolution criterion, both globally and locally. Using inhomogeneous diffusion and total variation regularization as examples for localized regularization schemes, we present an efficient method for locally adaptive image denoising. As expected, the smoothing parameter serves as an edge detector in this framework. Numerical examples illustrate the usefulness of our approach. We also present an application in confocal microscopy.

stat.ME↗

arXiv2009

Limiting Distributions for Sums of Independent Random Products

Let $\{X_{i,j}:(i,j)\in\mathbb N^2\}$ be a two-dimensional array of independent copies of a random variable $X$, and let $\{N_n\}_{n\in\mathbb N}$ be a sequence of natural numbers such that $\lim_{n\to\infty}e^{-cn}N_n=1$ for some $c>0$. Our main object of interest is the sum of independent random products $$Z_n=\sum_{i=1}^{N_n} \prod_{j=1}^{n}e^{X_{i,j}}.$$ It is shown that the limiting properties of $Z_n$, as $n\to\infty$, undergo phase transitions at two critical points $c=c_1$ and $c=c_2$. Namely, if $c>c_2$, then $Z_n$ satisfies the central limit theorem with the usual normalization, whereas for $c c_1$. If the random variable $X$ is Gaussian, we recover the results of Bovier, Kurkova, and Löwe [Fluctuations of the free energy in the REM and the $p$-spin SK models. Ann. Probab. 30(2002), 605-651].

math.PR↗

arXiv2009

Stationary max-stable fields associated to negative definite functions

Let $W_i,i\in{\mathbb{N}}$, be independent copies of a zero-mean Gaussian process $\{W(t),t\in{\mathbb{R}}^d\}$ with stationary increments and variance $σ^2(t)$. Independently of $W_i$, let $\sum_{i=1}^{\infty}δ_{U_i}$ 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 $\{V_i(\cdot),i\in{\mathbb{N}}\}$, where $V_i(t)=U_i+W_i(t)-σ^2(t)/2$, is translation invariant. In particular, the process $η(t)=\bigvee_{i=1}^{\infty}V_i(t)$ 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 $n\to\infty$ if and only if $W$ is a (nonisotropic) fractional Brownian motion on ${\mathbb{R}}^d$. Under suitable conditions on $W$, the process $η$ has a mixed moving maxima representation.

math.PR↗

arXiv2009

Extremes of Independent Gaussian Processes

For every $n\in\N$, let $X_{1n},..., X_{nn}$ be independent copies of a zero-mean Gaussian process $X_n=\{X_n(t), t\in T\}$. We describe all processes which can be obtained as limits, as $n\to\infty$, of the process $a_n(M_n-b_n)$, where $M_n(t)=\max_{i=1,...,n} X_{in}(t)$ and $a_n, b_n$ are normalizing constants. We also provide an analogous characterization for the limits of the process $a_nL_n$, where $L_n(t)=\min_{i=1,...,n} |X_{in}(t)|$.

math.PR↗

arXiv2009

$L^{2}$-spectral gaps, weak-reversible and very weak-reversible Markov chains

The theory of $L^2$-spectral gaps for reversible Markov chains has been studied by many authors. In this paper we consider positive recurrent general state space Markov chains with stationary transition probabilities. Replacing the assumption of reversibility by a less strong one, we still obtain a simple necessary and sufficient condition for the spectral gap property of the associated Markov operator in terms of isoperimetric constant. Moreover, we define a new sequence of isoperimetric constants which provides a necessary and sufficient condition for the existence of a spectral gap in a very general setting. Finally, these results are used to obtain simple sufficient conditions for the existence of a spectral gap in terms of the first and second order transition probabilities.

math.PR↗