Closure of some heavy-tailed distribution classes under random convolution*

In this paper, we consider the closure property of a random convolution $ \sum\nolimits_{n = 0}^\infty {{p_n}{F^*}^n} $ , where F is a heavy-tailed distribution on [0, ??), and p _n (n?=?0, 1, . . . ) are the local probabilities of a nonnegative integer-valued random variable. We obtain conditions under which the fact that distribution F belongs to the dominatedly varying-tailed class, long-tailed class, or to the intersection of these classes implies that $ \sum\nolimits_{n = 0}^\infty {{p_n}{F^*}^n} $ is in the same class.     

Random repeated quantum interactions and random invariant states

We consider a generalized model of repeated quantum interactions, where a system ${\mathcal{H}}$ is interacting in a random way with a sequence of independent quantum systems ${\mathcal{K}_n, n \geq 1}$ . Two types of randomness are studied in detail. One is provided by considering Haar-distributed unitaries to describe each interaction between ${\mathcal{H}}$ and ${\mathcal{K}_n}$ . The other involves random quantum states describing each copy ${\mathcal{K}_n}$ . In the limit of a large number of interactions, we present convergence results for the asymptotic state of ${\mathcal{H}}$ . This is achieved by studying spectral properties of (random) quantum channels which guarantee the existence of unique invariant states. Finally this allows to introduce a new physically motivated ensemble of random density matrices called the asymptotic induced ensemble.     

Universality properties of Gelfand–Tsetlin patterns

A standard Gelfand–Tsetlin pattern of depth n is a configuration of particles in ${\{1,\ldots,n\} \times \mathbb{R}}$ . For each ${r \in \{1, \ldots, n\}, \{r\} \times \mathbb{R}}$ is referred to as the rth level of the pattern. A standard Gelfand–Tsetlin pattern has exactly r particles on each level r, and particles on adjacent levels satisfy an interlacing constraint. Probability distributions on the set of Gelfand–Tsetlin patterns of depth n arise naturally as distributions of eigenvalue minor processes of random Hermitian matrices of size n. We consider such probability spaces when the distribution of the matrix is unitarily invariant, prove a determinantal structure for a broad subclass, and calculate the correlation kernel. In particular we consider the case where the eigenvalues of the random matrix are fixed. This corresponds to choosing uniformly from the set of Gelfand–Tsetlin patterns whose nth level is fixed at the eigenvalues of the matrix. Fixing ${q_n \in \{1,\ldots,n\}}$ , and letting n → ∞ under the assumption that ${\frac{q_n}n \to \alpha \in (0, 1)}$ and the empirical distribution of the particles on the nth level converges weakly, the asymptotic behaviour of particles on level q _n is relevant to free probability theory. Saddle point analysis is used to identify the set in which these particles behave asymptotically like a determinantal random point field with the Sine kernel.     

Exact value of the resistance exponent for four dimensional random walk trace

Let S be a simple random walk starting at the origin in ${\mathbb{Z}^{4}}$ . We consider ${{\mathcal G}=S[0,\infty)}$ to be a random subgraph of the integer lattice and assume that a resistance of unit 1 is put on each edge of the graph ${{\mathcal G}}$ . Let ${R_{{\mathcal G}}(0,S_{n})}$ be the effective resistance between the origin and S _n . We derive the exact value of the resistance exponent; more precisely, we prove that ${n^{-1}E(R_{{\mathcal G}}(0,S_{n}))\approx (\log n)^{-\frac{1}{2}}}$ . As an application, we obtain sharp heat kernel estimates for random walk on ${\mathcal G}$ at the quenched level. These results give the answer to the problem raised by Burdzy and Lawler (J Phys A Math Gen 23:L23–L28, 1990) in four dimensions.     

Mod-discrete expansions

In this paper, we consider approximating expansions for the distribution of integer valued random variables, in circumstances in which convergence in law (without normalization) cannot be expected. The setting is one in which the simplest approximation to the $n$ -th random variable  $X_n$ is by a particular member $R_n$ of a given family of distributions, whose variance increases with  $n$ . The basic assumption is that the ratio of the characteristic function of  $X_n$ to that of  $R_n$ converges to a limit in a prescribed fashion. Our results cover and extend a number of classical examples in probability, combinatorics and number theory.     

A quantitative Arrow theorem

Arrow’s Impossibility theorem states that any constitution which satisfies independence of irrelevant alternatives (IIA) and unanimity and is not a dictator has to be non-transitive. In this paper we study quantitative versions of Arrow theorem. Consider n voters who vote independently at random, each following the uniform distribution over the six rankings of three alternatives. Arrow’s theorem implies that any constitution which satisfies IIA and unanimity and is not a dictator has a probability of at least 6^?n for a non-transitive outcome. When n is large, 6^?n is a very small probability, and the question arises if for large number of voters it is possible to avoid paradoxes with probability close to 1. Here we give a negative answer to this question by proving that for every ${\epsilon > 0}$ , there exists a ${\delta = \delta(\epsilon) > 0}$ , which depends on ${\epsilon}$ only, such that for all n, and all constitutions on three alternatives, if the constitution satisfies:

• The IIA condition.
• For every pair of alternatives a, b, the probability that the constitution ranks a above b is at least ${\epsilon}$ .
• For every voter i, the probability that the social choice function agrees with a dictatorship on i at most ${1-\epsilon}$ .

Then the probability of a non-transitive outcome is at least δ. Our results generalize to any number k ≥ 3 of alternatives and to other distributions over the alternatives. We further derive a quantitative characterization of all social choice functions satisfying the IIA condition whose outcome is transitive with probability at least 1 ? δ. Our results provide a quantitative statement of Arrow theorem and its generalizations and strengthen results of Kalai and Keller who proved quantitative Arrow theorems for k?=?3 and for balanced constitutions only, i.e., for constitutions which satisfy for every pair of alternatives a, b, that the probability that the constitution ranks a above b is exactly 1/2. The main novel technical ingredient of our proof is the use of inverse-hypercontractivity to show that if the outcome is transitive with high probability then there are no two different voters who are pivotal with for two different pairwise preferences with non-negligible probability. Another important ingredient of the proof is the application of non-linear invariance to lower bound the probability of a paradox for constitutions where all voters have small probability for being pivotal.     

Boosted Surfaces: Synthesis of Meshes using Point Pair Generators as Curvature Operators in the 3D Conformal Model

This paper introduces a new technique for the formulation of parametric surfaces. Applying translation operations to tangent vectors ${n_{\circ} {\bf \upsilon}}$ results in null point pairs ${\tau}$ . We treat these null point pairs as surface and mesh curvature control points which can be interpolated and exponentiated to construct continuous topological transformations ${\mathcal{K}}$ of the form ${e^{{-}\frac{\tau}{2}}}$ 2. Some basic algorithms are proposed, including the boost which bends a line to a circle of curvature ${\kappa}$ , and the twisted boost which generates the Hopf fibration. We investigate methods to control curvature in two orthogonal directions u and v and examine a few distance-based and linear weighting techniques for synthesizing surface patches using multiple curvature control points. We consider the expressivity of the technique in manipulating meshes, and find that applying these rotors to mesh points provides a novel and computationally efficient method for creating boosted forms.     

Characteristic functions as distributional boundary values of analytic functions in tube domains

We propose necessary and sufficient conditions for a complex-valued function f on \( {{\mathbb{R}}^n} \) to be a characteristic function of a probability measure. Certain analytic extensions of f to tubular domains in \( {{\mathbb{C}}^n} \) are studied. In order to extend the class of functions under study, we also consider the case where f is a generalized function (distribution). The main result is given in terms of completely monotonic functions on convex cones in \( {{\mathbb{R}}^n} \) .     

Finite Diagonal Random Matrices

The goal of this article is to extend some results of Popescu (Probab. Theory Relat. Fields 144:179, 2009) in several directions. We establish the limiting spectral distribution (LSD) for r-diagonal matrices under reduced moment conditions compared to those required by Popescu. We also deal with the joint convergence of several sequences of such matrices. In particular, we show that there is a large class of such matrices where the joint limit is not free while the marginals are semicircular. We also consider matrices of the form $X_{n}X_{n}^{T}$ where X _n is a sequence of nonsymmetric r-diagonal random matrices and establish their limiting spectral distribution.     

Large Deviations for Proportions of Observations Which Fall in Random Sets Determined by Order Statistics

Let {X _n :n?≥?1} be independent random variables with common distribution function F and consider $K_{h:n}(D)=\sum_{j=1}^n1_{\{X_j-X_{h:n}\in D\}}$ , where h?∈?{1,...,n}, X _1:k ?≤???≤?X _k:k are the order statistics of the sample X ₁,...,X _k and D is some suitable Borel set of the real line. In this paper we prove that, if F is continuous and strictly increasing in the essential support of the distribution and if $\lim_{n\to\infty}\frac{h_n}{n}=\lambda$ for some λ?∈?[0,1], then $\{K_{h_n:n}(D)/n:n\geq 1\}$ satisfies the large deviation principle. As a by product we derive the large deviation principle for order statistics $\{X_{h_n:n}:n\geq 1\}$ . We also present results for the special case of Bernoulli distributed random variables with mean p?∈?(0,1), and we see that the large deviation principle holds only for p?≥?1/2. We discuss further almost sure convergence of $\{K_{h_n:n}(D)/n:n\geq 1\}$ and some related quantities.     

Generalized sub-Gaussian fractional Brownian motion queueing model

It is well known that often the one-dimensional distribution of a queue content is not Gaussian but its tails behave like a Gaussian. We propose to consider a general class of processes, namely the class of $\varphi $ -sub-Gaussian random processes, which is more general than the Gaussian one and includes non-Gaussian processes. The class of sub-Gaussian random processes contains Gaussian processes also and therefore is of special interest. In this paper we provide an estimate for the queue content distribution of a fluid queue fed by $N$ independent strictly $\varphi $ -sub-Gaussian generalized fractional Brownian motion input processes. We obtain an upper estimate of buffer overflow probability in a finite buffer system defined on any finite time interval $[a,b]$ or infinite interval $[0,\infty )$ . The derived estimate captures more accurately the performance of the queueing system for a wider-range of input processes.     

On slowdown and speedup of transient random walks in random environment

We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time n the particle is typically at a distance of order O(n ^κ ) from the origin, ${\kappa \in (0, 1)}$ . We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time n, the particle is at a distance of order ${O(n^{\nu_0})}$ from the origin, ${\nu_0 \in (0, \kappa)}$ ), and speedup (at time n, the particle is at a distance of order ${n^{\nu_1}}$ from the origin, ${\nu_1 \in (\kappa, 1)}$ ), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time n, the particle is located around (?n ^ν ), thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well.     

Lyapounov norms for random walks in low disorder and dimension greater than three

We consider a simple random walk on Z ^d , d > 3. We also consider a collection of i.i.d. positive and bounded random variables $\left(V_\omega(x)\right)_{x\in Z^d}$ , which will serve as a random potential. We study the annealed and quenched cost to perform long crossing in the random potential $-(\lambda+\beta V_\omega(x))$ , where λ is positive constant and β > 0 is small enough. These costs are measured by the Lyapounov norms. We prove the equality of the annealed and the quenched norm.     

The tail behaviour of a random sum of subexponential random variables and vectors

Let $\left\{ X,X_{i},i=1,2,...\right\} $ denote independent positive random variables having common distribution function (d.f.) F(x) and, independent of X, let ν denote an integer valued random variable. Using X ₀=0, the random sum Z=∑ _i=0 ^ν X _i has d.f. $G(x)=\sum_{n=0}^{\infty }\Pr\{\nu =n\}F^{n\ast }(x)$ where F ^n?(x) denotes the n-fold convolution of F with itself. If F is subexponential, Kesten’s bound states that for each ε>0 we can find a constant K such that the inequality $$ 1-F^{n\ast }(x)\leq K(1+\varepsilon )^{n}(1-F(x))\, , \qquad n\geq 1,x\geq 0 \, , $$ holds. When F is subexponential and E(1 +ε) ^ν <∞, it is a standard result in risk theory that G(x) satisfies $$ 1 - G{\left( x \right)} \sim E{\left( \nu \right)}{\left( {1 - F{\left( x \right)}} \right)},\,\,x \to \infty \,\,{\left( * \right)} $$ In this paper, we show that (*) holds under weaker assumptions on ν and under stronger conditions on F. Stam (Adv. Appl. Prob. 5:308–327, 1973) considered the case where $ \overline{F}(x)=1-F(x)$ is regularly varying with index –α. He proved that if α>1 and $E{\left( {\nu ^{{\alpha + \varepsilon }} } \right)} < \infty $ , then relation (*) holds. For 0<α<1, it is sufficient that Eν<∞. In this paper we consider the case where $\overline{F}(x)$ is an O-regularly varying subexponential function. If the lower Matuszewska index $\beta (\overline{F})<-1$ , then the condition ${\text{E}}{\left( {\nu ^{{{\left| {\beta {\left( {\overline{F} } \right)}} \right|} + 1 + \varepsilon }} } \right)} < \infty$ is sufficient for (*). If $\beta (\overline{F} )>-1$ , then again Eν<∞ is sufficient. The proofs of the results rely on deriving bounds for the ratio $\overline{F^{n\ast }}(x)/\overline{F} (x)$ . In the paper, we also consider (*) in the special case where X is a positive stable random variable or has a compound Poisson distribution derived from such a random variable and, in this case, we show that for n≥2, the ratio $\overline{F^{n\ast }}(x)/\overline{F}(x)\uparrow n$ as x↑∞. In Section 3 of the paper, we briefly discuss an extension of Kesten’s inequality. In the final section of the paper, we discuss a multivariate analogue of (*).     

Sharp Estimates for Turbulence in White-Forced Generalised Burgers Equation

We consider the non-homogeneous generalised Burgers equation $$\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} -\nu \frac{\partial^2 u}{\partial x^2} = \eta,\ t \geq 0,\ x \in S^1.$$ Here f is strongly convex and satisfies a growth condition, ν is small and positive, while η is a random forcing term, smooth in space and white in time. For any solution u of this equation we consider the quasi-stationary regime, corresponding to ${t \geq T_1}$ , where T ₁ depends only on f and on the distribution of η. We obtain sharp upper and lower bounds for Sobolev norms of u averaged in time and in ensemble. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence. All our bounds do not depend on the initial condition or on t for ${t \geq T_1}$ , and hold uniformly in ν. Estimates similar to some of our results have been obtained by Aurell, Frisch, Lutsko and Vergassola on a physical level of rigour; we use an argument from their article.     

Almost Exponential Maps and Integrability Results for a Class of Horizontally Regular Vector Fields

We consider a family ${\mathcal{H}}:= \{X_1, \dots, X_m\}$ of C ¹ vector fields in ? ⁿ and we discuss the associated ${\mathcal{H}}$ -orbits. Namely, we assume that our vector fields belong to a horizontal regularity class and we require that a suitable s-involutivity assumption holds. Then we show that any ${\mathcal{H}}$ -orbit ${\mathcal{O}}$ is a C ¹ immersed submanifold and it is an integral submanifold of the distribution generated by the family of all commutators up to length s. Our main tool is a class of almost exponential maps of which we discuss carefully some precise first order expansions.     

Normal approximation for a random elliptic equation

We consider solutions of an elliptic partial differential equation in \(\mathbb{R }^d\) with a stationary, random conductivity coefficient that is also periodic with period \(L\) . Boundary conditions on a square domain of width \(L\) are arranged so that the solution has a macroscopic unit gradient. We then consider the average flux that results from this imposed boundary condition. It is known that in the limit \(L \rightarrow \infty \) , this quantity converges to a deterministic constant, almost surely. Our main result is that the law of this random variable is very close to that of a normal random variable, if the domain size \(L\) is large. We quantify this approximation by an error estimate in total variation. The error estimate relies on a second order Poincaré inequality developed recently by Chatterjee.     

Complete bounded null curves immersed in {\mathbb {C}^3} and {\rm {SL}(2,\mathbb {C})}

We construct a simply connected complete bounded mean curvature one surface in the hyperbolic 3-space ${\mathcal {H}^3}$ . Such a surface in ${\mathcal {H}^3}$ can be lifted as a complete bounded null curve in ${\rm {SL}(2,\mathbb {C})}$ . Using a transformation between null curves in ${\mathbb {C}^3}$ and null curves in ${\rm {SL}(2,\mathbb {C})}$ , we are able to produce the first examples of complete bounded null curves in ${\mathbb {C}^3}$ . As an application, we can show the existence of a complete bounded minimal surface in ${\mathbb {R}^3}$ whose conjugate minimal surface is also bounded. Moreover, we can show the existence of a complete bounded immersed complex submanifold in ${\mathbb {C}^2}$ .     

Graphs with Average Degree Smaller Than {\frac {30}{11}} Burn Slowly

In this paper, we consider the following firefighter problem on a finite graph G =  (V, E). Suppose that a fire breaks out at a given vertex ${v \in V}$ . In each subsequent time unit, a firefighter protects one vertex which is not yet on fire, and then the fire spreads to all unprotected neighbours of the vertices on fire. The objective of the firefighter is to save as many vertices as possible. The surviving rate ${\rho(G)}$ of G is defined as the expected percentage of vertices that can be saved when a fire breaks out at a random vertex of G. Let ε >  0. We show that any graph G on n vertices with at most ${(\frac {15}{11} - \varepsilon)n}$ edges can be well protected, that is, ${\rho(G) > \frac {\varepsilon}{60} > 0}$ . Moreover, a construction of a random graph is proposed to show that the constant ${\frac {15}{11}}$ cannot be improved.     

Some diffusion processes associated with two parameter Poisson–Dirichlet distribution and Dirichlet process

The two parameter Poisson–Dirichlet distribution PD(α, θ) is the distribution of an infinite dimensional random discrete probability. It is a generalization of Kingman’s Poisson–Dirichlet distribution. The two parameter Dirichlet process ${\Pi_{\alpha,\theta,\nu_0}}$ is the law of a pure atomic random measure with masses following the two parameter Poisson–Dirichlet distribution. In this article we focus on the construction and the properties of the infinite dimensional symmetric diffusion processes with respective symmetric measures PD(α, θ) and ${\Pi_{\alpha,\theta,\nu_0}}$ . The methods used come from the theory of Dirichlet forms.