arXiv subjects

Zakhar Kabluchko

Publications and source records attributed to Zakhar Kabluchko.

At least 145 records · Page 8Linked to original sources

arXiv2016

Local universality for real roots of random trigonometric polynomials

Consider a random trigonometric polynomial $X_n: \mathbb R \to \mathbb R$ of the form $$ X_n(t) = \sum_{k=1}^n \left( ξ_k \sin (kt) + η_k \cos (kt)\right), $$ where $(ξ_1,η_1),(ξ_2,η_2),\ldots$ are independent identically distributed bivariate real random vectors with zero mean and unit covariance matrix. Let $(s_n)_{n\in\mathbb N}$ be any sequence of real numbers. We prove that as $n\to\infty$, the number of real zeros of $X_n$ in the interval $[s_n+a/n, s_n+ b/n]$ converges in distribution to the number of zeros in the interval $[a,b]$ of a stationary, zero-mean Gaussian process with correlation function $(\sin t)/t$. We also establish similar local universality results for the centered random vectors $(ξ_k,η_k)$ having an arbitrary covariance matrix or belonging to the domain of attraction of a two-dimensional $α$-stable law.

math.PR↗

arXiv2016

Renewal shot noise processes in the case of slowly varying tails

We investigate weak convergence of renewal shot noise processes in the case of slowly varying tails of the inter-shot times. We show that these processes, after an appropriate non-linear scaling, converge in the sense of finite-dimensional distributions to an inverse extremal process.

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arXiv2016

Mean width of regular polytopes and expected maxima of correlated Gaussian variables

An old conjecture states that among all simplices inscribed in the unit sphere the regular one has the maximal mean width. An equivalent formulation is that for any centered Gaussian vector $(ξ_1,\dots,ξ_n)$ satisfying $\mathbb Eξ_1^2= \dots =\mathbb Eξ_n^2=1$ one has $$ \mathbb E\,\max\{ξ_1,\dots,ξ_n\}\leq\sqrt{\frac{n}{n-1}}\, \mathbb E\,\max\{η_1,\dots,η_n\}, $$ where $η_1,η_2,\dots,$ are independent standard Gaussian variables. Using this probabilistic interpretation we derive an asymptotic version of the conjecture. We also show that the mean width of the regular simplex with $2n$ vertices is remarkably close to the mean width of the regular crosspolytope with the same number of vertices. Interpreted probabilistically, our result states that $$ 1\leq\frac{\mathbb E\,\max\{|η_1|,\dots,|η_n|\}}{\mathbb E\,\max\{η_1,\dots,η_{2n}\}} \leq\min\left\{\sqrt{\frac{2n}{2n-1}}, \, 1+\frac{C}{n\, \log n} \right\}, $$ where $C>0$ is an absolute constant. We also compute the higher moments of the projection length $W$ of the regular cube, simplex and crosspolytope onto a line with random direction, thus proving several formulas conjectured by S. Finch. Finally, we prove distributional limit theorems for the length of random projection as the dimension goes to $\infty$. In the case of the $n$-dimensional unit cube $Q_n$, we prove that $$ W_{Q_n} - \sqrt{\frac{2n}π} \overset{d}{\underset{n\to\infty}\longrightarrow} {\mathcal{N}} \left(0, \frac{π-3}π\right), $$ whereas for the simplex and the crosspolytope the limiting distributions are related to the Gumbel double exponential law.

math.PR↗

arXiv2016

Weak convergence of renewal shot noise processes in the case of slowly varying normalization

We investigate weak convergence of finite-dimensional distributions of a renewal shot noise process $(Y(t))_{t\geq 0}$ with deterministic response function $h$ and the shots occurring at the times $0 = S_0 < S_1 < S_2<\ldots$, where $(S_n)$ is a random walk with i.i.d.\ jumps. There has been an outbreak of recent activity around this topic. We are interested in one out of few cases which remained open: $h$ is regularly varying at $\infty$ of index $-1/2$ and the integral of $h^2$ is infinite. Assuming that $S_1$ has a moment of order $r>2$ we use a strong approximation argument to show that the random fluctuations of $Y(s)$ occur on the scale $s=t+g(t,u)$ for $u\in [0,1]$, as $t\to\infty$, and, on the level of finite-dimensional distributions, are well approximated by the sum of a Brownian motion and a Gaussian process with independent values (the two processes being independent). The scaling function $g$ above depends on the slowly varying factor of $h$. If, for instance, $\lim_{t\to\infty}t^{1/2}h(t)\in (0,\infty)$, then $g(t,u)=t^u$.

math.PR↗

arXiv2016

Inclusion-exclusion principles for convex hulls and the Euler relation

Consider $n$ points $X_1,\ldots,X_n$ in $\mathbb R^d$ and denote their convex hull by $Π$. We prove a number of inclusion-exclusion identities for the system of convex hulls $Π_I:=conv(X_i\colon i\in I)$, where $I$ ranges over all subsets of $\{1,\ldots,n\}$. For instance, denoting by $c_k(X)$ the number of $k$-element subcollections of $(X_1,\ldots,X_n)$ whose convex hull contains a point $X\in\mathbb R^d$, we prove that $$ c_1(X)-c_2(X)+c_3(X)-\ldots + (-1)^{n-1} c_n(X) = (-1)^{\dim Π} $$ for all $X$ in the relative interior of $Π$. This confirms a conjecture of R. Cowan [Adv. Appl. Probab., 39(3):630--644, 2007] who proved the above formula for almost all $X$. We establish similar results for the number of polytopes $Π_J$ containing a given polytope $Π_I$ as an $r$-dimensional face, thus proving another conjecture of R. Cowan [Discrete Comput. Geom., 43(2):209--220, 2010]. As a consequence, we derive inclusion-exclusion identities for the intrinsic volumes and the face numbers of the polytopes $Π_I$. The main tool in our proofs is a formula for the alternating sum of the face numbers of a convex polytope intersected by an affine subspace. This formula generalizes the classical Euler--Schläfli--Poincaré relation and is of independent interest.

math.PR↗

arXiv2016

Fractionally integrated inverse stable subordinators

A fractionally integrated inverse stable subordinator (FIISS) is the convolution of a power function and an inverse stable subordinator. We show that the FIISS is a scaling limit in the Skorokhod space of a renewal shot noise process with heavy-tailed, infinite mean `inter-shot' distribution and regularly varying response function. We prove local Hölder continuity of FIISS and a law of iterated logarithm for both small and large times.

math.PR↗

arXiv2016

Stochastic integral representations and classification of sum- and max-infinitely divisible processes

Introduced is the notion of minimality for spectral representations of sum- and max-infinitely divisible processes and it is shown that the minimal spectral representation on a Borel space exists and is unique. This fact is used to show that a stationary, stochastically continuous, sum- or max-i.d. random process on $\mathbb{R}^d$ can be generated by a measure-preserving flow on a $σ$-finite Borel measure space and that this flow is unique. This development makes it possible to extend the classification program of Rosiński (Ann. Probab. 23 (1995) 1163-1187) with a unified treatment of both sum- and max-infinitely divisible processes. As a particular case, a characterization of stationary, stochastically continuous, union-infinitely divisible random measurable subsets of $\mathbb{R}^d$ is obtained. Introduced and classified are several new max-i.d. random field models including fields of Penrose type and fields associated to Poisson line processes.

math.PR↗

arXiv2016

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

Let $(W_n(θ))_{n\in\mathbb N_0}$ be the Biggins martingale associated with a supercritical branching random walk and denote by $W_\infty(θ)$ its limit. Assuming essentially that the martingale $(W_n(2θ))_{n\in\mathbb N_0}$ is uniformly integrable and that $\text{Var} W_1(θ)$ is finite, we prove a functional central limit theorem for the tail process $(W_\infty(θ) - W_{n+r}(θ))_{r\in\mathbb N_0}$ and a law of the iterated logarithm for $W_\infty(θ)-W_n(θ)$, as $n\to\infty$.

math.PR↗

arXiv2015

A characterization of the normal distribution using stationary max-stable processes

Consider the max-stable process $η(t) = \max_{i\in\mathbb N} U_i \rm{e}^{\langle X_i, t\rangle - κ(t)}$, $t\in\mathbb{R}^d$, where $\{U_i, i\in\mathbb{N}\}$ are points of the Poisson process with intensity $u^{-2}\rm{d} u$ on $(0,\infty)$, $X_i$, $i\in\mathbb{N}$, are independent copies of a random $d$-variate vector $X$ (that are independent of the Poisson process), and $κ: \mathbb{R}^d \to \mathbb{R}$ is a function. We show that the process $η$ is stationary if and only if $X$ has multivariate normal distribution and $κ(t)-κ(0)$ is the cumulant generating function of $X$. In this case, $η$ is a max-stable process introduced by R. L. Smith.

math.PR↗

arXiv2015

Max-stable processes and stationary systems of Lévy particles

We study stationary max-stable processes $\{η(t)\colon t\in\mathbb R\}$ admitting a representation of the form $η(t)=\max_{i\in\mathbb N}(U_i+ Y_i(t))$, where $\sum_{i=1}^{\infty} δ_{U_i}$ is a Poisson point process on $\mathbb R$ with intensity ${\rm e}^{-u} {\rm d} u$, and $Y_1,Y_2,\ldots$ are i.i.d.\ copies of a process $\{Y(t)\colon t\in\mathbb R\}$ obtained by running a Lévy process for positive $t$ and a dual Lévy process for negative $t$. We give a general construction of such Lévy-Brown-Resnick processes, where the restrictions of $Y$ to the positive and negative half-axes are Lévy processes with random birth and killing times. We show that these max-stable processes appear as limits of suitably normalized pointwise maxima of the form $M_n(t)=\max_{i=1,\ldots,n} ξ_i(s_n+t)$, where $ξ_1,ξ_2,\ldots$ are i.i.d.\ Lévy processes and $s_n$ is a sequence such that $s_n\sim c \log n$ with $c>0$. Also, we consider maxima of the form $\max_{i=1,\ldots,n} Z_i(t/\log n)$, where $Z_1,Z_2,\ldots$ are i.i.d.\ Ornstein--Uhlenbeck processes driven by an $α$-stable noise with skewness parameter $β=-1$. After a linear normalization, we again obtain limiting max-stable processes of the above form. This gives a generalization of the results of Brown and Resnick [Extreme values of independent stochastic processes, J.\ Appl.\ Probab., 14 (1977), pp.\ 732--739] to the totally skewed $α$-stable case.

math.PR↗

arXiv2015

Least Energy Approximation for Processes with Stationary Increments

A function $f=f_T$ is called least energy approximation to a function $B$ on the interval $[0,T]$ with penalty $Q$ if it solves the variational problem $$ \int_0^T \left[ f'(t)^2 + Q(f(t)-B(t)) \right] dt \searrow \min. $$ For quadratic penalty the least energy approximation can be found explicitly. If $B$ is a random process with stationary increments, then on large intervals $f_T$ also is close to a process of the same class and the relation between the corresponding spectral measures can be found. We show that in a long run (when $T\to \infty$) the expectation of energy of optimal approximation per unit of time converges to some limit which we compute explicitly. For Gaussian and Lévy processes we complete this result with almost sure and $L^1$ convergence. As an example, the asymptotic expression of approximation energy is found for fractional Brownian motion.

math.PR↗

arXiv2015

A functional central limit theorem for branching random walks, almost sure weak convergence, and applications to random trees

Let $W_{\infty}(β)$ be the limit of the Biggins martingale $W_n(β)$ associated to a supercritical branching random walk with mean number of offspring $m$. We prove a functional central limit theorem stating that as $n\to\infty$ the process $$ D_n(u):= m^{\frac 12 n} \left(W_{\infty}\left(\frac{u}{\sqrt n}\right) - W_{n}\left(\frac{u}{\sqrt n}\right) \right) $$ converges weakly, on a suitable space of analytic functions, to a Gaussian random analytic function with random variance. Using this result we prove central limit theorems for the total path length of random trees. In the setting of binary search trees, we recover a recent result of R. Neininger [Refined Quicksort Asymptotics, Rand. Struct. and Alg., to appear], but we also prove a similar theorem for uniform random recursive trees. Moreover, we replace weak convergence in Neininger's theorem by the almost sure weak (a.s.w.) convergence of probability transition kernels. In the case of binary search trees, our result states that $$ L\left\{\sqrt{\frac{n}{2\log n}} \left(EPL_{\infty} - \frac{EPL_n-2n\log n}{n}\right)\Bigg | G_{n}\right\} \to \{ω\mapsto N_{0,1}\}, \quad \text{a.s.w.},$$ where $EPL_n$ is the external path length of a binary search tree $X_n$ with $n$ vertices, $EPL_{\infty}$ is the limit of the Régnier martingale, and $L(\,\cdot\, |G_n)$ denotes the conditional distribution w.r.t. the $σ$-algebra $G_n$ generated by $X_1,\ldots,X_n$. A.s.w. convergence is stronger than weak and even stable convergence. We prove several basic properties of the a.s.w. convergence and study a number of further examples in which the a.s.w. convergence appears naturally. These include the classical central limit theorem for Galton-Watson processes and the Pólya urn.

math.PR↗

arXiv2015

Maxima of independent, non-identically distributed Gaussian vectors

Let $X_{i,n},n\in \mathbb{N},1\leq i\leq n$, be a triangular array of independent $\mathbb{R}^d$-valued Gaussian random vectors with correlation matrices $Σ_{i,n}$. We give necessary conditions under which the row-wise maxima converge to some max-stable distribution which generalizes the class of Hüsler-Reiss distributions. In the bivariate case, the conditions will also be sufficient. Using these results, new models for bivariate extremes are derived explicitly. Moreover, we define a new class of stationary, max-stable processes as max-mixtures of Brown-Resnick processes. As an application, we show that these processes realize a large set of extremal correlation functions, a natural dependence measure for max-stable processes. This set includes all functions $ψ(\sqrt{γ(h)}),h\in \mathbb{R}^d$, where $ψ$ is a completely monotone function and $γ$ is an arbitrary variogram.

math.PR↗

arXiv2014

Asymptotic distribution of complex zeros of random analytic functions

Let $ξ_0,ξ_1,\ldots$ be independent identically distributed complex- valued random variables such that $\mathbb{E}\log(1+|ξ_0|)<\infty$. We consider random analytic functions of the form \[\mathbf{G}_n(z)=\sum_{k=0}^{\infty}ξ_kf_{k,n}z^k,\] where $f_{k,n}$ are deterministic complex coefficients. Let $μ_n$ be the random measure counting the complex zeros of $\mathbf{G}_n$ according to their multiplicities. Assuming essentially that $-\frac{1}{n}\log f_{[tn],n}\to u(t)$ as $n\to\infty$, where $u(t)$ is some function, we show that the measure $\frac{1}{n}μ_n$ converges in probability to some deterministic measure $μ$ which is characterized in terms of the Legendre-Fenchel transform of $u$. The limiting measure $μ$ 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↗

arXiv2014

Intrinsic volumes of Sobolev balls with applications to Brownian convex hulls

A formula due to Sudakov relates the first intrinsic volume of a convex set in a Hilbert space to the maximum of the isonormal Gaussian process over this set. Using this formula we compute the first intrinsic volumes of infinite-dimensional convex compact sets including unit balls with respect to Sobolev-type seminorms and ellipsoids in the Hilbert space. We relate the distribution of the random one-dimensional projections of these sets to the distributions $S_1,S_2,C_1,C_2$ studied by Biane, Pitman, Yor [Bull.\ AMS 38 (2001)]. We show that the $k$-th intrinsic volume of the set of all functions on $[0,1]$ which have Lipschitz constant bounded by $1$ and which vanish at $0$ (respectively, which have vanishing integral) is given by $$ V_k = \frac{π^{k/2}}{Γ\left(\frac 32 k +1 \right)}, \text{ respectively } V_k = \frac{π^{(k+1)/2}}{2Γ\left(\frac 32 k +\frac 32\right)}. $$ This is related to the results of Gao and Vitale [Discrete Comput. Geom. 26 (2001), Elect. Comm. Probab. 8 (2003)] who considered a similar question for functions with a restriction on the total variation instead of the Lipschitz constant. Using the results of Gao and Vitale we give a new proof of the formula for the expected volume of the convex hull of the $d$-dimensional Brownian motion which is due to Eldan [Elect. J. Probab., to appear]. Additionally, we prove an analogue of Eldan's result for the Brownian bridge. Similarly, we show that the results on the intrinsic volumes of the Lipschitz balls can be translated into formulae for the expected volumes of zonoids (Aumann integrals) generated by the Brownian motion and the Brownian bridge. Also, these results have discrete versions for Gaussian random walks and bridges. Our proofs exploit Sudakov's and Tsirelson's theorems which establish a connection between the intrinsic volumes and the isonormal Gaussian process.

math.PR↗

arXiv2014

Limiting distribution for the maximal standardized increment of a random walk

Let $X_1,X_2,...$ be independent identically distributed random variables with $\mathbb E X_k=0$, $\mathrm{Var} X_k=1$. Suppose that $φ(t):=\log \mathbb E e^{t X_k}<\infty$ for all $t>-σ_0$ and some $σ_0>0$. Let $S_k=X_1+...+X_k$ and $S_0=0$. We are interested in the limiting distribution of the multiscale scan statistic $$ M_n=\max_{0\leq i 0$. In this case, we show that the main contribution to $M_n$ comes from the intervals $(i,j)$ having length $l:=j-i$ of order $a(\log n)^{p}$, $a>0$, where $p=q/(q-2)$ and $q\in{3,4,...}$ is the order of the first non-vanishing cumulant of $X_1$ (not counting the variance). In the logarithmic case we assume that the function $ψ(t):=2φ(t)/t^2$ attains its maximum $m_*>1$ at some unique point $t=t_*\in (0,\infty)$. In this case, we show that the main contribution to $M_n$ comes from the intervals $(i,j)$ of length $d_*\log n+a\sqrt{\log n}$, $a\in\mathbb R$, where $d_*=1/φ(t_*)>0$. In the sublogarithmic case we assume that the tail of $X_k$ is heavier than $\exp\{-x^{2-\varepsilon}\}$, for some $\varepsilon>0$. In this case, the main contribution to $M_n$ comes from the intervals of length $o(\log n)$ and in fact, under regularity assumptions, from the intervals of length $1$. In the remaining, fourth case, the $X_k$'s are Gaussian. This case has been studied earlier in the literature. The main contribution comes from intervals of length $a\log n$, $a>0$. We argue that our results cover most interesting distributions with light tails.

math.PR↗

arXiv2014

Generalized Random Energy Model at Complex Temperatures

Motivated by the Lee--Yang approach to phase transitions, we study the partition function of the Generalized Random Energy Model (GREM) at complex inverse temperature $β$. We compute the limiting log-partition function and describe the fluctuations of the partition function. For the GREM with $d$ levels, in total, there are $\frac 12 (d+1)(d+2)$ phases, each of which can symbolically be encoded as $G^{d_1}F^{d_2}E^{d_3}$ with $d_1,d_2,d_3\in\mathbb{N}_0$ such that $d_1+d_2+d_3=d$. In phase $G^{d_1}F^{d_2}E^{d_3}$, the first $d_1$ levels (counting from the root of the GREM tree) are in the glassy phase (G), the next $d_2$ levels are dominated by fluctuations (F), and the last $d_3$ levels are dominated by the expectation (E). Only the phases of the form $G^{d_1}E^{d_3}$ intersect the real $β$ axis. We describe the limiting distribution of the zeros of the partition function in the complex $β$ plane (= Fisher zeros). It turns out that the complex zeros densely touch the positive real axis at $d$ points at which the GREM is known to undergo phase transitions. Our results confirm rigorously and considerably extend the replica-method predictions from the physics literature.

math.PR↗

arXiv2013

Roots of random polynomials whose coefficients have logarithmic tails

It has been shown by Ibragimov and Zaporozhets [In Prokhorov and Contemporary Probability Theory (2013) Springer] that the complex roots of a random polynomial $G_n(z)=\sum_{k=0}^nξ_kz^k$ with i.i.d. coefficients $ξ_0,\ldots,ξ_n$ concentrate a.s. near the unit circle as $n\to\infty$ if and only if ${\mathbb{E}\log_+}|ξ_0|<\infty$. We study the transition from concentration to deconcentration of roots by considering coefficients with tails behaving like $L({\log}|t|)({\log}|t|)^{-α}$ as $t\to\infty$, where $α\geq0$, and $L$ is a slowly varying function. Under this assumption, the structure of complex and real roots of $G_n$ is described in terms of the least concave majorant of the Poisson point process on $[0,1]\times (0,\infty)$ with intensity $αv^{-(α+1)}\,du\,dv$.

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