Asymptotic behaviour of multiperiodic functions scaled by Pisot numbers

Let β > 1 be a Pisot number andg be a positive Hölder continuous function with period one and g(0) = 1. The multiperiodic functionG(ξ)=Π _n=0 ^∞ g(ξ/βⁿ) is studied and the asymptotic behaviour ofI _G(T) = ∫ ₀ ^T G(ξ)dξ investigated. We prove that the limit of logI(T)/ logT exists asT tends to infinity. We also provide a method to calculate this limit for the caseg(ξ) = cos² 2πξ, corresponding to the Fourier transform of the Bernoulli convolution associated to the golden number (or some of its generalizations).     

Ideals and free Pairs in the semigroup β ℤ

We prove that the equations ξ+x=mξ+y, x+ξ=y+mξ have no solutions in the semigroup β ? for every free ultrafilter ξ and every integer m∈0, 1. We study semigroups generated by the ultrafilters ξ, mξ. For left maximal idempotents, we prove a reduced hypothesis about elements of finite order in β ?.     

On strong chains of uncountable functions

For functionsf,g:ω₁ → ω₁, where ω₁ is the first uncountable cardinal, we write thatf?g if and only if {ξ ∈ ω₁ :f(ξ)≥g(ξ)} is finite. We prove the consistency of the existence of a well-ordered increasing ?-chain of length ω₁₂, solving a problem of A. Hajnal. The methods previously developed by us involveforcing with side conditions in morasses which is a variation on Todorcevic'sforcing with models as side conditions. The paper is self-contained and requires from the reader knowledge of Kunen's textbook and some basic experience with proper forcing and elementary submodels.     

Regular Semigroups Each of Whose Least Completely Simple Congruence Classes Has a Greatest Element

A regular semigroup S is termed locally F-regular, if each class of the least completely simple congruence ξ contains a greatest element with respect to the natural partial order. It is shown that each locally F-regular semigroup S admits an embedding into a semidirect product of a band by S/ξ. Further, if ξ satisfies the additional property that for each s ∈ S and each inverse (sξ)′ of sξ in S/ξ the set (sξ)′ ∩ V(s) is not empty, we represent S both as a Rees matrix semigroup over an F-regular semigroup as well as a certain subsemigroup of a restricted semidirect product of a band by S/ξ.     

Boundary proximity of SLE

This paper examines how close the chordal SLE _κ curve gets to the real line asymptotically far away from its starting point. In particular, when κ ? (0, 4), it is shown that if β > β _κ  := 1/(8/κ ? 2), then the intersection of the SLE _κ curve with the graph of the function y = x/(log x) ^β , x > e, is a.s. bounded, while it is a.s. unbounded if β = β _κ . The critical SLE₄ curve a.s. intersects the graph of $y=x^{{-({\rm log\,log\,x})}^{\alpha}}, x > e^e$ , x > e ^e , in an unbounded set if α ≤ 1, but not if α > 1. Under a very mild regularity assumption on the function y(x), we give a necessary and sufficient integrability condition for the intersection of the SLE _κ path with the graph of y to be unbounded. When the intersection is bounded a.s., we provide an estimate for the probability that the SLE _κ path hits the graph of y. We also prove that the Hausdorff dimension of the intersection set of the SLE _κ curve and the real axis is 2 ? 8/κ when 4 < κ < 8.     

On strong limit theorems concerning delayed sums of a random sequence

Let (ξ _n ) _n∈N be a sequence of arbitrarily dependent random variables. In this paper, a generalized strong limit theorem of the delayed average of (ξ _n ) _n∈N is investigated, then some limit theorems for arbitrary information sources follow. As corollaries, some known results are generalized.     

带跳的分数维Brown 运动幂变差的渐近行为

本文研究了X_t = B^H_t + ξ_t 现实幂变差的渐近理论, B^H 为Hurst 指数为H∈(0,1) 的分数维Brown 运动,ξ为与B^H独立的非Gauss Lévy 过程, 我们给出了其大数定律, 以及经适当中心化的中 心极限定理, 这些结果将为处理具有长期记忆跳过程的统计问题提供理论基础.     

On the singularity of Quillen metrics

Let π: X → S be a holomorphic map from a compact Kähler manifold (X,g _X ) to a compact Riemann surface S. Let Σ_π be the critical locus of π and let Δ  =  π(Σ_π) be the discriminant locus. Let (ξ, h _ξ) be a holomorphic Hermitian vector bundle on X. We determine the singularity of the Quillen metric on det Rπ_*ξ near Δ with respect to g _X | _TX/S and h _ξ.     

Conditional limit theorems for critical continuous-state branching processes

We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism ψ(λ) = λ1+αL(1/λ), where α∈ [0, 1] and L is slowly varying at ∞. We prove that if α∈(0, 1], there are norming constants Qt→ 0(as t ↑ +∞) such that for every x 0, Px(QtXt∈·| Xt 0)converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of ψ at0. We give a conditional limit theorem for the case α = 0. The limit theorems we obtain in this paper allow infinite variance of the branching process.     

Iterated Palm Conditioning and Some Slivnyak-Type Theorems for Cox and Cluster Processes

A huge literature is devoted to extensions and ramifications of Slivnyak’s celebrated theorem—the fact that a point process on a Borel space is Poisson if and only if its reduced Palm distributions agree almost everywhere with the original distribution. Here we prove some multivariate versions of the mentioned property for randomizations and Cox processes. This leads to Slivnyak-type theorems for certain Cox cluster processes, arising naturally both in exchangeability theory and in the context of superprocesses. Our proofs are based on a general iteration principle, which enables us to calculate Palm measures via a preliminary conditioning.     

Limite non visqueuse pour le système de Navier–Stokes dans un espace critique

In a recent paper, Vishik proved the global well-posedness of the two-dimensional Euler equation in the critical Besov space B_2,1². In the present paper we prove that Navier–Stokes system is globally well-posed in B_2,1², with uniform estimates on the viscosity. We prove also a global result of inviscid limit. The convergence rate in L² is of order ν. To cite this article: T. Hmidi, S. Keraani, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Normalisation d'un processus de branchement critique dans un environnement aléatoire

Let (Z_n) be a critical branching process in an independent and identically distributed (i.i.d.) random environment. For each fixed environment ω, let C_n=E^ω[Z_n∣Z_n>0] be the conditional expectation of Z_n given Z_n>0. We prove an analogue of Yaglom's law: as n→∞, the conditional law of Z_n/C_n, conditional on Z_n>0, converges to a non-degenerate law on [0,∞). We give also an analogue of Kolmogorov's law, as well as a local limit theorem for the semi-group of probability generating functions. To cite this article: Y. Guivarc'h et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Multivariate cauchy distributions as locally gaussian distributions

In the present paper, we propose a definition of locally Gaussian probability distributions of random vectors based on the linearization of their conditional quantiles. We prove that the Cauchy distribution inR ⁿ is locally Gaussian and give explicit formulas for the vectors of expectations and covariance matrices of locally Gaussian approximations. We show that locally Gaussian approximations with different dimensionalities are in some sense compatible: all of them have equal corresponding correlation coefficients. For the Cauchy distribution in a Hilbert space we prove a limit theorem on the convergence of squared finite-dimensional conditional quantiles to the stable Lévy distribution. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models. Part II. Eger, Hungary, 1994.     

Central limit theorems for power variation of Gaussian integral processes with jumps

This paper presents limit theorems for realized power variation of processes of the form Xt=t0φsdGs+ξt as the sampling frequency within a fixed interval increases to infinity.Here G is a Gaussian process with stationary increments,ξis a purely non-Gaussian L′evy process independent from G,andφis a stochastic process ensuring that the integral is well defined as a pathwise Riemann-Stieltjes integral.We obtain the central limit theorems for the case that both the continuous term and the jump term are presented simultaneously in the law of large numbers.     

On Solvability of Boundary Value Problems for Kinetic Operator-Differential Equations

We study solvability of boundary value problems for the so-called kinetic operator-differential equations of the form B(t)u _t ?L(t)u = f, where L(t) and B(t) are families of linear operators defined in a complex Hilbert space E. We do not assume that the operator B is invertible and that the spectrum of the pencil L ?λ B is included into one of the half-planes Re λ < a or Re λ > a \({(a\in {\mathbb{R}})}\) . Under certain conditions on the above operators, we prove several existence and uniqueness theorems and study smoothness questions in weighted Sobolev spaces for solutions.     

Local probabilities for random walks conditioned to stay positive

Let S ₀ = 0, {S _n , n ≥ 1} be a random walk generated by a sequence of i.i.d. random variables X ₁, X ₂, . . . and let $\tau ^{-}={\rm min} \{ n \geq 1:S_{n}\leq 0 \}$ and $\tau ^{+}={\rm min}\{n\geq1:S_{n} > 0\} $ . Assuming that the distribution of X ₁ belongs to the domain of attraction of an α-stable law we study the asymptotic behavior, as ${n\rightarrow \infty }$ , of the local probabilities ${\bf P}{(\tau ^{\pm }=n)}$ and prove the Gnedenko and Stone type conditional local limit theorems for the probabilities ${\bf P}{(S_{n} \in [x,x+\Delta )|\tau^{-} > n)}$ with fixed Δ and ${x=x(n)\in (0,\infty )}$ .     

Some Estimates for the Norm of the Bergman Projection on Besov Spaces

In this paper we obtain an estimate of the norm of the Bergman projection from L ^p (D, dλ) onto the Besov space B _p , 1 < p < + ∞. The result is asymptotically sharp when p → + ∞. Further for the case P : L ¹(D, dλ) → B ₁, we consider some weak type inequalities with the corresponding spaces.     

On the KŁR conjecture in random graphs

The K?R conjecture of Kohayakawa, ?uczak, and Rödl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph G _n,p , for sufficiently large p:= p(n), satisfy an embedding lemma which complements the sparse regularity lemma of Kohayakawa and Rödl. We prove a variant of this conjecture which is sufficient for most known applications to random graphs. In particular, our result implies a number of recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems. We also discuss several further applications.