Papers by Zakhar Kabluchko
Statistics & Probability Letters, 2016
We investigate weak convergence of finite-dimensional distributions of renewal shot noise process... more We investigate weak convergence of finite-dimensional distributions of renewal shot noise process (Y (t)) t≥0 with deterministic response function h and the shots occurring at the times 0 = S 0 < S 1 < S 2 < . . ., 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 ∞ 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 ∈ [0, 1] 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→∞ t 1/2 h(t) ∈ (0, ∞), then g(t, u) = t u .
Stochastic Processes and their Applications, 2016
Electronic Journal of Probability, 2016
Rmp, 2003
The review of the representation theory of deformations of the CCR is presented. The faithfulness... more The review of the representation theory of deformations of the CCR is presented. The faithfulness of the Fock representation of q-CCR, twisted CCR and quon CCR is discussed. The more general deformation of CCR is presented. The K0 and K1 groups of the twisted CCR algebra are calculated.
Let $\xi_0,\xi_1,...$ be independent identically distributed (i.i.d.) random variables such that ... more Let $\xi_0,\xi_1,...$ be independent identically distributed (i.i.d.) random variables such that $\E \log (1+|\xi_0|)<\infty$. We consider random analytic functions of the form $$ G_n(z)=\sum_{k=0}^{\infty} \xi_k f_{k,n} z^k, $$ where $f_{k,n}$ are deterministic complex coefficients. Let $\nu_n$ 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 $\nu_n$ 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 $\xi_k$'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.
Consider a branching random walk on Z in discrete time. Denote by Ln(k) the number of particles a... more Consider a branching random walk on Z in discrete time. Denote by Ln(k) the number of particles at site k ∈ Z at time n ∈ N 0 . By the profile of the branching random walk (at time n) we mean the function k → Ln(k). We establish the following asymptotic expansion of Ln(k), as n → ∞:
We consider a system of independent branching random walks on $\R$ which start off a Poisson poin... more We consider a system of independent branching random walks on $\R$ which start off a Poisson point process with intensity of the form $e_{\lambda}(du)=e^{-\lambda u}du$, where $\lambda\in\R$ is chosen in such a way that the overall intensity of particles is preserved. Denote by $\chi$ the cluster distribution and let $\phi$ be the log-Laplace transform of the intensity of $\chi$. If $\lambda\phi'(\lambda)>0$, 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 $\Pi_{e_{\lambda}}^{\chi}$ with intensity $e_{\lambda}$. If $\lambda\phi'(\lambda)<0$, 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 $\chi$ as mixtures of the point processes $\Pi_{ce_{\lambda}}^{\chi}$ over $c>0$ and $\lambda\in K_{\text{st}}$, where $K_{\text{st}}=\{\lambda\in\R: \phi(\lambda)=0, \lambda\phi'(\lambda)>0\}$.
A formula due to Sudakov relates the first intrinsic volume of a convex set in a Hilbert space to... more 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 ]. 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
The theory of $L^2$-spectral gaps for reversible Markov chains has been studied by many authors. ... more 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.
Journal of Theoretical Probability, 2015
A function f = f T is called least energy approximation to a function B on the interval [0, T ] w... more 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 T 0 2010
A formula due to Sudakov relates the first intrinsic volume of a convex set in a Hilbert space to... more 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 ]. 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
ABSTRACT We prove that the appropriately normalized maximum of the Gaussian $1/f^{\alpha}$-noise ... more ABSTRACT We prove that the appropriately normalized maximum of the Gaussian $1/f^{\alpha}$-noise with $\alpha&lt;1$ converges in distribution to the Gumbel double-exponential law. Comment: 7 pages
Let $W_i,i\in{\mathbb{N}}$, be independent copies of a zero-mean Gaussian process $\{W(t),t\in{\m... more 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 $\sigma^2(t)$. Independently of $W_i$, let $\sum_{i=1}^{\infty}\delta_{U_i}$ 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 $V_i(t)=U_i+W_i(t)-\sigma^2(t)/2$, is translation invariant. In particular, the process $\eta(t)=\bigvee_{i=1}^{\infty}V_i(t)$ is a stationary
Journal of Fluid Mechanics, 2012
ABSTRACT The partition function of the random energy model at inverse temperature $\beta$ is a su... more ABSTRACT The partition function of the random energy model at inverse temperature $\beta$ is a sum of random exponentials $Z_N(\beta)=\sum_{k=1}^N \exp(\beta \sqrt{n} X_k)$, 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 $\beta$. 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 $\beta$, 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.
Let $\xi_i$, $i\in \mathbb {N}$, be independent copies of a L\'{e}vy process $\{\xi(t),t\geq0... more Let $\xi_i$, $i\in \mathbb {N}$, be independent copies of a L\'{e}vy process $\{\xi(t),t\geq0\}$. Motivated by the results obtained previously in the context of the random energy model, we prove functional limit theorems for the process \[Z_N(t)=\sum_{i=1}^N\mathrm{e}^{\xi_i(s_N+t)}\] 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 $s_N$.
Let {Xi,j : (i, j) ∈ N 2 } be a two-dimensional array of independent copies of a random variable ... more Let {Xi,j : (i, j) ∈ N 2 } be a two-dimensional array of independent copies of a random variable X, and let {Nn} n∈N be a sequence of natural numbers such that limn→∞ e −cn Nn = 1 for some c > 0. Our main object of interest is the sum of independent random products
Let {X i , i = 1, 2, . . .} be i.i.d. standard gaussian variables. Let S n = X 1 + . . . + X n be... more Let {X i , i = 1, 2, . . .} be i.i.d. standard gaussian variables. Let S n = X 1 + . . . + X n be the sequence of partial sums and
A formula due to Sudakov relates the first intrinsic volume of a convex set in a Hilbert space to... more 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 ]. 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
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Papers by Zakhar Kabluchko