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Summary Fubini theorem is an essential tool for the analysis of high-dimensional space [8], [2], [3], a theorem about the multiple integral and iterated integral. The author has been working on formalizing Fubini’s theorem over the past few years [4], [6] in the Mizar system [7], [1]. As a result, Fubini’s theorem (30) was proved in complete form by this article.

Under natural assumptions about the measure spaces, the topological size as well as the algebraic size of the family of measurable real functions on the product measure space satisfying or not the conclusion of the Fubini theorem are analyzed. In particular, a (nontrivial) nonlineable set is provided.

We show that repulsive random variables can yield Monte Carlo methods with faster convergence rates than the typical $N^{-1/2}$, where $N$ is the number of integrand evaluations. More precisely, we propose stochastic numerical quadratures involving determinantal point processes associated with multivariate orthogonal polynomials, and we obtain root mean square errors that decrease as $N^{-(1+1/d)/2}$, where $d$ is the dimension of the ambient space. First, we prove a central limit theorem (CLT) for the linear statistics of a class of determinantal point processes, when the reference measure is a product measure supported on a hypercube, which satisfies the Nevai-class regularity condition, a result which may be of independent interest. Next, we introduce a Monte Carlo method based on these determinantal point processes, and prove a CLT with explicit limiting variance for the quadrature error, when the reference measure satisfies a stronger regularity condition. As a corollary, by taking a specific reference measure and using a construction similar to importance sampling, we obtain a general Monte Carlo method, which applies to any measure with continuously derivable density. Loosely speaking, our method can be interpreted as a stochastic counterpart to Gaussian quadrature, which, at the price of some convergence rate, is easily generalizable to any dimension and has a more explicit error term., Comment: 58 pages, 6 figures. To appear in Annals of Applied Probability

The main result of this paper is a bi-parameter Tb theorem for Littlewood–Paley g-function, where b is a tensor product of two pseudo-accretive function. Instead of the doubling measure, we work with a product measure μ = μn × μm, where the measures μn and μm are only assumed to be upper doubling. The main techniques of the proof include a bi-parameter b-adapted Haar function decomposition and an averaging identity over good double Whitney regions. Moreover, the non-homogeneous analysis and probabilistic methods are used again.

Quasi-Monte Carlo (QMC) methods are equal weight quadrature rules to approximate integrals over the unit cube with respect to the uniform measure. In this paper we discuss QMC integration with respect to general product measures defined on an arbitrary hyperrectangle. We only require that the cumulative distribution function is invertible. We develop a worst-case error bound and study the dependence of the error on the number of points and the dimension for digital nets and sequences as well as polynomial lattice point sets, which are mapped to the domain using the inverse cumulative distribution function. We do not require any smoothness properties of the probability density function and the worst-case error does not depend on the particular choice of density function and its smoothness. The component-by-component construction of polynomial lattice rules is based on a criterion which depends only on the size of the hyperrectangle but is otherwise independent of the product measure.

Summary The purpose of this article is to show Fubini’s theorem on measure [16], [4], [7], [15], [18]. Some theorems have the possibility of slight generalization, but we have priority to avoid the complexity of the description. First of all, for the product measure constructed in [14], we show some theorems. Then we introduce the section which plays an important role in Fubini’s theorem, and prove the relevant proposition. Finally we show Fubini’s theorem on measure.

The distinguishing number D(G) of a graph G is the minimum number of colors needed to color the vertices of G such that the coloring is preserved only by the trivial automorphism. In this paper we improve results about the distinguishing number of Cartesian products of finite and infinite graphs by removing restrictions to prime or relatively prime factors.

International audience; We consider a generalization of the Bauer maximum principle. We work with tensorial products of convex measures sets, that are non necessarily compact but generated by their extreme points. We show that the maximum of a quasi-convex lower semicontinuous function on this product space is reached on the tensorial product of finite mixtures of extreme points. Our work is an extension of the Bauer maximum principle in three different aspects. First, we only assume that the objective functional is quasi-convex. Secondly, the optimization is performed over a space built as a product of measures sets. Finally, the usual compactness assumption is replaced with the existence of an integral representation on the extreme points. We focus on product of two different types of measures sets, called the moment class and the unimodal moment class. The elements of these classes are probability measures (respectively unimodal probability measures) satisfying generalized moment constraints. We show that an integral representation on the extreme points stands for such spaces and that it extends to their tensorial product. We give several applications of the Theorem, going from robust Bayesian analysis to the optimization of a quantile of a computer code output.

[EN] Given s-finite measure spaces ( 1, 1, mu 1) and ( 2, 2, mu 2), we consider Banach spaces X1(mu 1) and X2(mu 2), consisting of L0(mu 1) and L0(mu 2) measurable functions respectively, and study when the completion of the simple tensors in the projective tensor product X1(mu 1). p X2(mu 2) is continuously included in the metric space of measurable functions L0(mu 1. mu 2). In particular, we prove that the elements of the completion of the projective tensor product of L p-spaces are measurable functions with respect to the product measure. Assuming certain conditions, we finally showthat given a bounded linear operator T : X1(mu 1). p X2(mu 2). E (where E is a Banach space), a norm can be found for T to be bounded, which is ` minimal' with respect to a given property (2-rectangularity). The same technique may work for the case of n-spaces., J. M. Calabuig and M. Fernandez-Unzueta were supported by Ministerio de Economia, Industria y Competitividad (Spain) under project MTM2014-53009-P. M. Fernandez-Unzueta was also suported by CONACyT 284110. F. Galaz-Fontes was supported by Ministerio de Ciencia e Innovacion (Spain) and FEDER under project MTM2009-14483-C02-01. E. A. Sanchez Perez was supported by Ministerio de Economia, Industria y Competitividad (Spain) and FEDER under project MTM2016-77054-C2-1-P.

The main purpose of this paper is to establish a boundedness result for strong maximal functions with respect to certain non-doubling measures in $\mathbb{R}^{n}$ . More precisely, let $d\mu(x_{1}, \ldots, x_{n})=d\mu_{1}(x_{1})\cdots d\mu_{n}(x_{n})$ be a product measure which is not necessarily doubling in $\mathbb{R}^{n}$ (only assuming $d\mu_{i}$ is doubling on $\mathbb{R}$ for $i=2, \ldots, n$ ), and let ω be a nonnegative and locally integral function such that $\omega _{i}(\cdot)=\omega(x_{1}, \ldots, x_{i-1}, \cdot, x_{i+1}, \ldots, x_{n})$ is in $A_{\infty}^{1}(d\mu_{i})$ uniformly in $x_{1}, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{n}$ for each $i=1, \ldots, n-1$ , let $d\nu=\omega \,d\mu$ , $\nu(E)=\int_{E} \omega(y)\,d\mu(y)$ , and $M_{\omega \,d\mu}^{n}$ be the strong maximal function defined by $$M_{\omega \,d\mu}^{n} f(x)=\sup_{x\in R\in\mathcal{R}} \frac{1}{\nu (R)} \int_{R} \bigl\vert f(y) \bigr\vert \omega(y)\,d\mu(y), $$ where $\mathcal{R}$ is the collection of rectangles with sides parallel to the coordinate axes in $\mathbb{R}^{n}$ . Then we show that $M_{\omega \,d\mu}^{n} $ is bounded on $L^{p}_{\omega \,d\mu}(\mathbb{R}^{n})$ for $1< p

Summary In this article we formalize in Mizar [5] product pre-measure on product sets of measurable sets. Although there are some approaches to construct product measure [22], [6], [9], [21], [25], we start it from σ-measure because existence of σ-measure on any semialgebras has been proved in [15]. In this approach, we use some theorems for integrals.

For each of the four standard products of graphs, namely the Cartesian, the strong, the direct, and the lexicographic product, we characterize when a nontrivial product of two graphs is chordal, interval, or circular-arc, respectively.

This paper is concerned with a class of singular stable-like Dirichlet forms on R d , which are generated by d independent copies of a one-dimensional symmetric α-stable process, and whose Levy jump kernel measure is concentrated on the union of the coordinate axes. Explicit and sharp criteria for Poincare inequality, super Poincare inequality and weak Poincare inequality of such singular Dirichlet forms are presented. When the reference measure is a product measure on R d , we also consider the entropy inequality for the associated Dirichlet forms, which is similar to the log-Sobolev inequality for local Dirichlet forms, and enjoys the tensorisation property.

The measurability problems can arise in Gaussian and empirical processes underlying nonparametric and semiparametric problems critical in modern statistics and machine learning. The usual applications are high-dimensional data in random fields. To ensure the validity of recovering a true original complex object from its observed images treated in a product space that appears in existing high-dimensional data applications widely arising in physical, biological, and social sciences and engineering, we begin with the results on the σ-algebra of the original space Ω induced by a given map f T mapping from Ω to a product measure space ( Ω T , F T , μ T ) . We study the certain measurability issues on Ω with the induced measure μ from the image measure μ T of the product space by establishing the outer measure ν ⁎ restricted to the trace σ-algebra. We further investigate the measure μ on Ω under the map f T to obtain the conditions for the identifiability equality μ T = μ ∘ f T − 1 to hold so that a practical and rigorous foundation is provided for the nontrivial applications to work meaningfully.

© 2014 Elsevier B.V. All rights reserved. In arbitrage-free but incomplete markets, the equivalent martingale measure Q for pricing traded assets is not uniquely determined. A possible approach when it comes to choosing a particular pricing measure is to consider the one that is 'closest' to the physical probability measure P, where closeness is measured in terms of relative entropy. In this paper, we determine the minimal entropy martingale measure in a market where securities are traded with payoffs depending on two types of risks, which we will call financial and actuarial risks, respectively. In case only purely financial and purely actuarial securities are traded, we prove that financial and actuarial risks are independent under the physical measure if and only if these risks are independent under the entropy measure. Moreover, in such a market the entropy measure of the combined financial-actuarial world is the product measure of the entropy measures of the financial and the actuarial subworlds, respectively. ispartof: Journal of Computational and Applied Mathematics vol:282 pages:111-133 status: published

Graph products and their structural properties have been studied extensively by many researchers. We investigate the Laplacian eigenvalues and eigenvectors of the product graphs for the four standard products, namely, the Cartesian product, the direct product, the strong product and the lexicographic product. A complete characterization of Laplacian spectrum of the Cartesian product of two graphs has been done by Merris. We give an explicit complete characterization of the Laplacian spectrum of the lexicographic product of two graphs using the Laplacian spectra of the factors. For the other two products, we describe the complete spectrum of the product graphs in some particular cases. We supply some new results relating to the algebraic connectivity of the product graphs. We describe the characteristic sets for the Cartesian product and for the lexicographic product of two graphs. As an application we construct new classes of Laplacian integral graphs.

This article investigates selection of variables in high-dimension from a non-parametric regression model. In many concrete situations, we are concerned with estimating a non-parametric regression function f that may depend on a large number p of inputs variables. Unlike standard procedures, we do not assume that f belongs to a class of regular functions (Holder, Sobolev, ...), yet we assume that f is a square-integrable function with respect to a known product measure. Furthermore, observe that, in some situations, only a small number s of the coordinates actually affects f in an additive manner. In this context, we prove that, with only 𝒪(s log p ) random evaluations of f , one can find which are the relevant input variables with overwhelming probability. Our proposed method is an unconstrained l 1 -minimization procedure based on the Sobol’s method. One step of this procedure relies on support recovery using l 1 -minimization and thresholding. More precisely, we use a thresholded-LASSO to faithfully uncover the significant input variables. In this frame, we prove that one can relax the mutual incoherence property (known to require 𝒪(s 2 log p ) observations) and still ensure faithful recovery from 𝒪(s α log p ) observations for any 1 ≤ α ≤ 2.

This paper is concerned with dissipativity theory and robust performance analysis and design of discrete-time stochastic systems driven by statistically uncertain random noise. The uncertainty is quantified by the conditional relative entropy of the actual probability law of the noise with respect to a nominal product measure corresponding to a white noise sequence. We discuss a balance equation, dissipation inequality, and superadditivity property for the corresponding conditional relative entropy supply as a function of time. The problem of minimizing the supply, required to drive the system between given state distributions over a specified time horizon, is considered. Such variational problems, involving entropy and probabilistic boundary conditions, are known in the literature as Schrodinger bridge problems. In application to control systems, the minimum required conditional relative entropy supply characterizes the robustness of the system with respect to a statistically uncertain random noise. We o...

In this paper, we introduce the p-adic Hardy type operator and obtain its sharp bound on the p-adic Lebesgue product spaces. Meanwhile, an analogous result is computed for the p-adic Lebesgue product spaces with power weights. In addition, we characterize a sufficient and necessary condition which ensures that the weighted p-adic Hardy type operator is bounded on the p-adic Lebesgue product spaces. Furthermore, the p-adic weighted Hardy-Cesàro operator is also obtained.

We investigate the thermodynamic formalism for recurrent potentials on group extensions of countable Markov shifts. Our main result characterizes recurrent potentials depending only on the base space, in terms of the existence of a conservative product measure and a homomorphism from the group into the multiplicative group of real numbers. We deduce that, for a recurrent potential depending only on the base space, the group is necessarily amenable. Moreover, we give equivalent conditions for the base pressure and the skew product pressure to coincide. Finally, we apply our results to analyse the Poincaré series of Kleinian groups and the cogrowth of group presentations.

We study aspects of the analytic foundations of integration and closely related problems for functions of infinitely many variables x"1,x"2,[email protected]?D. The setting is based on a reproducing kernel k for functions on D, a family of non-negative weights @c"u, where u varies over all finite subsets of N, and a probability measure @r on D. We consider the weighted superposition [email protected]?"[email protected]"uk"u of finite tensor products k"u of k. Under mild assumptions we show that K is a reproducing kernel on a properly chosen domain in the sequence space D^N, and that the reproducing kernel Hilbert space H(K) is the orthogonal sum of the spaces H(@c"uk"u). Integration on H(K) can be defined in two ways, via a canonical representer or with respect to the product measure @r^N on D^N. We relate both approaches and provide sufficient conditions for the two approaches to coincide.

We present an approach that allows one to bound the largest and smallest singular values of an N × n random matrix with iid rows, distributed according to a measure on Rn that is supported in a relatively small ball and linear functionals are uniformly bounded in Lp for some p > 8, in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of 1 ± c √ n/N not only in the case of the above mentioned measure, but also when the measure is log-concave or when it a product measure of iid random variables with “heavy tails”.

Let n , k ∈ N with n ≥ 2 and 1 ≤ k n . Given a positive function γ ∈ C ∞ ( R n − k ) we form the Riemannian metric g ˜ on R n associated to the differential expression d s 2 = | d x ′ | 2 + γ ( x ′ ) 2 | d y | 2 where we write R n ∋ x = ( x ′ , y ) with x ′ ∈ R n − k and y ∈ R k . Let ν be a log-convex measure on R k with smooth density and μ the product measure μ : = ρ L n − k ⊗ ν on R n where ρ ∈ C ( R n − k ) is a positive function. We obtain a Polya–Szego inequality of the form ∫ R n f ( u , j ( ∇ g ˜ u ) ) d μ ≥ ∫ R n f ( u s , j ( ∇ g ˜ u s ) ) d μ for Sobolev functions u where the operation ⋅ s refers to the ( k , n ) -Steiner symmetrisation with respect to ν. The gradient operator ∇ g ˜ is associated to the metric g ˜ and the mapping j may be seen as interpolating between the tangent space at x and R n . The nonnegative integrand f is continuous and convex in the gradient variable and satisfies some additional hypotheses. As an application we derive a Polya–Szego inequality in the hyperbolic plane that takes the above form.

Classical results of Shepp and Feldman give a criterion for a product measure which is a countable power of a measure on ℝ with positive density to be equivalent to its shift by any vector in l2. In this work a similar problem is studied for shifts of a measure by vectors in l q 1 ⩽ q < 2.

In the paper we present two results for measures on projections in a W*-algebra of type I2. First, it is shown that, for any such measure m, there exists a Hilbert-valued orthogonal vector measure µ such that ‖µ(p)‖2 = m(p) for every projection p. In view of J. Hamhalter’s result (Proc. Amer. Math. Soc., 110 (1990), 803–806), this means that the above assertion is valid for an arbitrary W*-algebra. Secondly, a construction of a product measure on projections in a W*-algebra of type I2 (an analogue of the product measure in classical Lebesgue theory) is proposed.

We prove the existence of a successful coupling for $n$ particles in the symmetric inclusion process. As a consequence we characterize the ergodic measures with finite moments, and obtain sufficient conditions for a measure to converge in the course of time to an invariant product measure., Comment: 19 pages, 2 figures

We initiate the study of general neighborhood growth dynamics on two dimensional Hamming graphs. The decision to add a point is made by counting the currently occupied points on the horizontal and the vertical line through it, and checking whether the pair of counts lies outside a fixed Young diagram. We focus on two related extremal quantities. The first is the size of the smallest set that eventually occupies the entire plane. The second is the minimum of an energy-entropy functional that comes from the scaling of the probability of eventual full occupation versus the density of the initial product measure within a rectangle. We demonstrate the existence of this scaling and study these quantities for large Young diagrams., 55 pages

Liggett and Steif (2006) proved that, for the supercritical contact process on certain graphs, the upper invariant measure stochastically dominates an i.i.d.\ Bernoulli product measure. In particular, they proved this for $\mathbb{Z}^d$ and (for infection rate sufficiently large) $d$-ary homogeneous trees $T_d$. In this paper we prove some space-time versions of their results. We do this by combining their methods with specific properties of the contact process and general correlation inequalities. One of our main results concerns the contact process on $T_d$ with $d\geq2$. We show that, for large infection rate, there exists a subset $\Delta$ of the vertices of $T_d$, containing a "positive fraction" of all the vertices of $T_d$, such that the following holds: The contact process on $T_d$ observed on $\Delta$ stochastically dominates an independent spin-flip process. (This is known to be false for the contact process on graphs having subexponential growth.) We further prove that the supercritical contact process on $\mathbb{Z}^d$ observed on certain $d$-dimensional space-time slabs stochastically dominates an i.i.d.\ Bernoulli product measure, from which we conclude strong mixing properties important in the study of certain random walks in random environment., Comment: Minor modifications/corrections, for Random Structures & Algorithms

Under the assumption that the underlying measure is a non-negative Radon measure which only satisfies some growth condition and may not be doubling, we define the product of functions in the regular BMO and the atomic block H1 in the sense of distribution, and show that this product may be split into two parts, one in L1 and the other in some Hardy-Orlicz space.

Consider a class of skew product transformations consisting of an ergodic or a periodic transformation on a probability space (M, B, m) in the base and a semigroup of transformations on another probability space (W,F,P) in the fibre. Under suitable mixing conditions for the fibre transformation, we show that the properties ergodicity, weakly mixing, and strongly mixing are passed on from the base transformation to the skew product (with respect to the product measure). We derive ergodic theorems with respect to the skew product on the product space. The main aim of this paper is to establish uniform convergence with respect to the base variable for the series of ergodic averages of a function F on the product of the two probability spaces along the orbits of such a skew product. Assuming a certain growth condition for the coupling function, a strong mixing condition on the fibre transformation, and continuity andintegrability conditions for F, we prove uniform convergence in the base and L^p(P)-convergence in the fibre. Under an equicontinuity assumption on F we further show P-almost sure convergence in the fibre. Our work has an application in information theory: It implies convergence of the averages of functions on random fields restricted to parts of stair climbing patterns defined by a direction., 16 pages, removing tex-commands from abstract, correction in affiliation

Continuing in the vein of a recently developed generalization of continuum thermomechanics, in this paper we extend fracture mechanics and beam mechanics to materials described by fractional integrals involving D, d and R. By introducing a product measure instead of a Riesz measure, so as to ensure that the mechanical approach to continuum mechanics is consistent with the energetic approach, specific forms of continuum-type equations are derived. On this basis we study the energy aspects of fracture and, as an example, a Timoshenko beam made of a fractal material; the local form of elastodynamic equations of that beam is derived. In particular, we review the crack driving force G stemming from the Griffith fracture criterion in fractal media, considering either dead-load or fixed-grip conditions and the effects of ensemble averaging over random fractal materials.

In this paper, the definition of intuitionistic fuzzy inner product is given. By virtue of this definition, some convergence theorems, Schwarts inequality and the orthogonal concept for intuitionistic fuzzy inner product spaces are established and introduced. Moreover the relationship between this kind of spaces and intuitionistic fuzzy normed spaces is considered.

We prove the quantum version of an ergodic result of H. Furstenberg relative to noninvariant measures. The natural setting will be the case of the "quantum diagonal measure" relative to the product measure. Even if in all the interesting situations such diagonal measures are neither invariant nor normal with respect to the corresponding product ones, we still provide an ergodic theorem for them, generalizing the classical case. As a natural application, we are able to prove the entangled ergodic theorem in some interesting situations out of the known ones, that is when the unitary is not almost periodic, or when the involved operators are not compact.

In this article, we describe a wide class of non-Volterra quadratic stochastic operators using N. Ganikhadjaev's construction of quadratic stochastic operators. By the construction these operators depend on a probability measure μ being defined on the set of all configurations which are given on a graph G. We show that if μ is the product of probability measures being defined on each maximal connected subgraphs of G then corresponding non-Volterra operator can be reduced to m number (where m is the number of maximal connected subgraphs of G) of Volterra operators defined on the maximal connected subgraphs. Our result allows to study a wide class of non-Volterra operators in the framework of the well known theory of Volterra quadratic stochastic operators.

A new operation on graphs is introduced and some of its properties are studied. We call it hierarchical product, because of the strong (connectedness) hierarchy of the vertices in the resulting graphs. In fact, the obtained graphs turn out to be subgraphs of the cartesian product of the corresponding factors. Some well-known properties of the cartesian product, such as reduced mean distance and diameter, simple routing algorithms and some optimal communication protocols are inherited by the hierarchical product. We also address the study of some algebraic properties of the hierarchical product of two or more graphs. In particular, the spectrum of the binary hypertree Tm (which is the hierarchical product of several copies of the complete graph on two vertices) is fully characterized; turning out to be an interesting example of graph with all its eigenvalues distinct. Finally, some natural generalizations of the hierarchic product are proposed.

We consider a positive distribution Φ such that Φ defines a probability measure μ=μΦ on the dual of some real nuclear Frechet space. A large deviation principle is proved for the family {μn,n≥1} where μn denotes the image measure of the product measure μΦn under the empirical distribution function Ln. Here the rate function I is defined on the space ℱ′θ(N′)+ and agrees with the relative entropy function \(\widetilde{H}(\Psi/\Phi)\) . As an application, we cite the Gibbs conditioning principle which describes the limiting behaviour as n tends to infinity of the law of k tagged particles Y1,…,Yk under the constraint that LnY belongs to some subset A0.

It is shown that one-sided interpolation of compactness property is possible by any method on the class of pseudo-lattices, i.e., spaces of measurable functions, where the operators PDf = f χD are uniformly bounded. Analogous results are shown for sequence spaces, ordered quasinormed Abelian groups and even extended to abstract Banach couples, satisfying some weak approximation hypothesis. Mathematics subject classification (2000): 46B70, 46E30.

For a probability measure R on a product of two probability spaces that is absolutely continuous with respect to the product measure we prove the existence of liftings subordinated to a regular conditional probability and the existence of a lifting for R with lifted sections which satisfies in addition a rectangle formula. These results improve essentially some of the results from the former work of the authors [W. Strauss, N.D. Macheras, K. Musial, Splitting of liftings in products of probability spaces, Ann. Probab. 32 (2004) 2389–2408], by weakening considerably the assumptions and by presenting more direct and shorter proofs. In comparison with [W. Strauss, N.D. Macheras, K. Musial, Splitting of liftings in products of probability spaces, Ann. Probab. 32 (2004) 2389–2408] it is crucial for applications intended that we can now prescribe one of the factor liftings completely freely. We demonstrate the latter by applications to τ -additive measures, transfer of strong liftings, and stochastic processes.

In large random economies with heterogeneous agents, a standard stochastic framework presumes a random macro state, combined with idiosyncratic micro shocks. This can be formally represented by a random process consisting of a continuum of random variables that are conditionally independent given the macro state. However, this process satisfies a standard joint measurability condition only if there is essentially no idiosyncratic risk at all. Based on iteratively complete product measure spaces, we characterize the validity of the standard stochastic framework via Monte Carlo simulation as well as event-wise measurable conditional probabilities. These general characterizations also allow us to strengthen some earlier results related to exchangeability and independence.

Let $N$ be a finite set, let $p \in (0,1)$, and let $N_p$ denote a random binomial subset of $N$ where every element of $N$ is taken to belong to the subset independently with probability $p$ . This defines a product measure $\mu_p$ on the power set of $N$, where for $\mathcal{A} \subseteq 2^N$ $\mu_p(\mathcal{A}) := Pr[N_p \in \mathcal{A}]$. In this paper we study upward-closed families $\mathcal{A}$ for which all minimal sets in $\mathcal{A}$ have size at most $k$, for some positive integer $k$. We prove that for such a family $\mu_p(\mathcal{A}) / p^k $ is a decreasing function, which implies a uniform bound on the coarseness of the thresholds of such families. We also prove a structure theorem which enables to identify in $\mathcal{A}$ either a substantial subfamily $\mathcal{A}_0$ for which the first moment method gives a good approximation of its measure, or a subfamily which can be well approximated by a family with all minimal sets of size strictly smaller than $k$. Finally, we relate the (fractional) expectation threshold and the probability threshold of such a family, using duality of linear programming. This is related to the threshold conjecture of Kahn and Kalai.

In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality \[ \mu(\lambda A + (1-\lambda)B)^{1/n} \geq \lambda \mu(A)^{1/n} + (1-\lambda)\mu(B)^{1/n} \] holds true for an unconditional product measure $\mu$ with decreasing density and a pair of unconditional convex bodies $A,B \subset \mathbb{R}^n$. We also show that the above inequality is true for any unconditional $\log$-concave measure $\mu$ and unconditional convex bodies $A,B \subset \mathbb{R}^n$. Finally, we prove that the inequality is true for a symmetric $\log$-concave measure $\mu$ and a pair of symmetric convex sets $A,B \subset \mathbb{R}^2$, which, in particular, settles two-dimensional case of the conjecture for Gaussian measure proposed by R. Gardner and the fourth named author. In addition, we deduce the $1/n$-concavity of the parallel volume $t \mapsto \mu(A+tB)$, Brunn's type theorem and certain analogues of Minkowski first inequality., Comment: 16 pages, 1 figure

We consider the nearest neighbor asymmetric exclusion process on Z , in which particles jump with probability p ( 1 ) to the right and p ( - 1 ) to the left. Let q = p ( 1 ) / p ( - 1 ) and denote by ν q an ergodic component of the reversible Bernoulli product measure which places a particle at x with probability q x / ( 1 + q x ) . It is well known that under some hypotheses on a local function V , ( 1 / t ) ∫ 0 t V ( η s ) d s converges to a normal distribution with variance σ 2 = σ 2 ( q ) , which depends on q . We prove in this article that σ 2 ( q ) is a C ∞ function of q on ( 0 , 1 ) .

For the packing measure of the Cartesian product of the middle third Cantor set with itself, the exact value P log 3 4 (C×C)=4 log 3 4 has been obtained.

Let [Formula: see text] be the class of all algebraic probability spaces. A "natural product" is, by definition, a map [Formula: see text] which is required to satisfy all the canonical axioms of Ben Ghorbal and Schürmann for "universal product" except for the commutativity axiom. We show that there exist only five natural products, namely tensor product, free product, Boolean product, monotone product and anti-monotone product. This means that, in a sense, there exist only five universal notions of stochastic independence in noncommutative probability theory.

We study the left action α of a Cartan subgroup on the space X = G/Γ, where Γ is a lattice in a simple split connected Lie group G of rank n > 1. Let μ be an α-invariant measure on X. We give several conditions using entropy and conditional measures, each of which characterizes the Haar measure on X. Furthermore, we show that the conditional measure on the foliation of unstable manifolds has the structure of a product measure. The main new element compared to the previous work on this subject is the use of noncommutativity of root foliations to establish rigidity of invariant measures. © 2003 Wiley Periodicals, Inc.

In modelling economic decision processes with risk and uncertainty one has to update the measures and condition the functions given new information. Non-additive measures (also called Choquet capacities) proved to be an important tool for modelling uncertainty. Here, for functions on a finite set Ω, a comprehensive approach to conditioning is made. It builds on conditional expectation for probability measures in representing the monotone measure ν as a min of belief functions below ν and the latter as a max of additive measures in the core of the respective belief function. The same method is applied, too, for defining the product of monotone measures.

A notion of "quasi-universal product" for algebraic probability spaces is introduced as a generalization of Speicher's "universal product". It is proved that there exist only five quasi-universal products, namely, tensor product, free product, Boolean product, monotone product and anti-monotone product. This result means that, in a sense, there exist only five independences which have nice properties of "associativity" and "(quasi-)universality".

Statistical equilibrium lattice models of coherent structures in geostrophic turbulence, formulated by discretizing the governing Hamiltonian continuum dynamics, are analyzed. The first set of results concern large deviation principles (LDP's) for a spatially coarse-grained process with respect to either the canonical and/or the microcanonical formulation of the model. These principles are derived from a basic LDP for the coarse-grained process with respect to product measure, which in turn depends on Cramer's Theorem. The rate functions for the LDP's ´ give rise to variational principles that determine the equilibrium solutions of the Hamiltonian equations. The second set of results addresses the equivalence or nonequivalence of the microcanonical and canonical ensembles. In particular, necessary and sufficient conditions for a correspondence between microcanonical equilibria and canonical equilibria are established in terms of the concavity of the microcanonical entropy. A complete characterization of equivalence of ensembles is deduced by elementary methods of convex analysis. The mathematical results proved in this paper complement the physical reasoning and numerical computations given in a companion paper, where it is argued that the statistical equilibrium model defined by a prior distribution on potential vorticity fluctuations and microcanonical conditions on total energy and circulation is natural from the perspective of geophysical applications. Large Deviation Principle, Cramer's Theorem, Coarse Graining

The usual concept of asymptotic independence, as discussed in the context of extreme value theory, requires the distribution of the coordinatewise sample maxima under suitable centering and scaling to converge to a product measure. However, this definition is too broad to conclude anything interesting about the tail behavior of the product of two random variables that are asymptotically independent. Here we introduce a new concept of asymptotic independence which allows us to study the tail behavior of products. We carefully discuss equivalent formulations of asymptotic independence. We then use the concept in the study of a network traffic model. The usual infinite source Poisson network model assumes that sources begin data transmissions at Poisson time points and continue for random lengths of time. It is assumed that the data transmissions proceed at a constant, nonrandom rate over the entire length of the transmission. However, analysis of network data suggests that the transmission rate is also random with a regularly varying tail. So, we modify the usual model to allow transmission sources to transmit at a random rate over the length of the transmission. We assume that the rate and the time have finite mean, infinite variance and possess asymptotic independence, as defined in the paper. We finally prove a limit theorem for the input process showing that the centered cumulative process under a suitable scaling converges to a totally skewed stable Lévy motion in the sense of finite-dimensional distributions.

In order to model a random time evolution, the canonical procedure is to construct probability measures on product spaces. Roughly speaking, the first step is to take a probability measure that models the initial distribution. In the second step, on a different probability space, the distribution after one time step is modeled. Then in each subsequent step, on a further probability space, the random state in the next time step given the full history is modeled. On a formal level, we consider products of probability spaces and Markov kernels between such spaces. Finally, the Ionescu-Tulcea theorem shows that the whole procedure can be realized on a single infinite product space. Furthermore, Kolmogorov’s extension theorem shows that a similar construction can be performed even if the time set is not discrete.