O. Khorunzhiy, J. -F. Marckert
Let D be a Dyck path chosen uniformly from the set of Dyck paths with 2n steps. We prove that the sequence of the exponential moments of the maximum of D normalized by the square root of n converges in the limit of infinite n, and therefore is bounded uniformly in n. This result justifies corresponding assumption used to prove certain estimates of high moments of large random matrices.
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We consider a model for a queue in which only a fixed number $N$ of customers can join. Each customer joins the queue independently at an exponentially distributed time. Assuming further that the service times are independent and follow an exponential distribution, this system can be described as a two-dimensional Markov process on a finite triangular region $\mathfrak S$ of the square lattice. We interpret the resulting random walk on $\mathfrak S$ as a Dyck path that is weighted according to some state-dependent transition probabilities that are constant along one axis, but are rather genera…
In this manuscript we consider the set of Dyck paths equipped with the uniform measure, and we study the statistical properties of a deformation of the observable "area below the Dyck path" as the size $N$ of the path goes to infinity. The deformation under analysis is apparently new: while usually the area is constructed as the sum of the heights of the steps of the Dyck path, here we regard it as the sum of the lengths of the connected horizontal slices under the path, and we deform it by applying to the lengths of the slices a positive regular function $ω(\ell)$ such that $ω(\ell) \sim \ell…
We establish necessary and sufficient conditions for the uniform integrability of the stochastic exponential E(M).
We study ergodic properties of stochastic dissipative systems with additive noise. We show that the system is uniformly exponentially ergodic provided the growth of nonlinearity at infinity is faster than linear. The abstract result is applied to the stochastic reaction diffusion equation in $\mathbb R^d$ with $d\le 3$.
We consider a rectangular grid induced by the south-west records from the planar Poisson point process in $R^2_+$. A random symmetry property of the matrix whose entries are the areas of tiles of the grid implies cute multivariate distributional identities for certain rational functions of independent exponential and uniform random variables.
Generally the convergence rate in exponential ergodicity $λ$ is an upper bound for the convergence rate $κ$ in uniform ergodicity for a Markov process, that is $λ\geqslantκ$. In this paper, we prove that $κ\geqslant \inf \{lambda,1/M_H\}$, where $M_H$ is a uniform bound on the moment of the hitting time to a "compact" set $H$. In the case where $M_H$ can be made arbitrarily small for $H$ large enough, we obtain that $λ=κ$. The general results are applied to Markov chains, diffusion processes and solutions to SDEs driven by symmetric stable processes.
We prove a Berry-Esseen theorem, a local central limit theorem and (local) large and (global) moderate deviations principles for i.i.d. (uniformly) random non-uniformly expanding or hyperbolic maps with exponential first return times. Using existing results the problem is reduced to certain random (Young) tower extensions, which is the main focus of this paper. On the random towers we will obtain our results using contraction properties of random complex equivariant cones with respect to the complex Hilbert projective metric.
Uniform exponential convergence of SAA with AMIS and asymptotics of its optimal value
本文研究了带有自适应多重重要性采样的样本平均近似的一致指数收敛性及最优值渐近性，利用有界鞅差集中不等式获得新指数收敛速率，并基于泛函中心极限定理与Delta定理证明最优值渐近性。
Let S_i be a random walk with standard exponential increments. We call \sum_{i=1}^k S_i its k-step area. The random variable V = \inf_{k \ge 1} \frac{2}{k(k+1)} \sum_{i=1}^k S_i plays important role in the study of so-called one-dimensional sticky particles model. We find the distribution of V and prove that P(V > t) = \sqrt{1-t} exp(-t/2) for t in [0,1]. We also show that the variables \min_{1 \le k \le n} \frac{2n}{k(k+1)} \sum_{i=1}^k U_{i, n} converge in distribution to V. Here U_{i, n} are the order statistics of n i.i.d. random variables uniformly distributed on [0,1].
In the first part of this paper, we give a useful criterion for uniform integrability of exponential martingales in the context of Markov processes. The condition of this criterion is easy to verify and is, in general, much weaker than the commonly used Novikov's condition. In the second part of this paper, we present a new approach to the study of spectral bounds of Feynman-Kac semigroups for a large class of symmetric Markov processes. We first establish criteria for the $L^p$-independence of spectral bounds for Feynman-Kac semigroups generated by continuous additive functionals, using gauge…
We study the mixing and dissipation properties of the advection-diffusion equation with diffusivity $0 < κ\ll 1$ and advection by a class of random velocity fields on $\mathbb T^d$, $d=\{2,3\}$, including solutions of the 2D Navier-Stokes equations forced by sufficiently regular-in-space, non-degenerate white-in-time noise. We prove that the solution almost surely mixes exponentially fast uniformly in the diffusivity $κ$. Namely, that there is a deterministic, exponential rate (independent of $κ$) such that all mean-zero $H^1$ initial data decays exponentially fast in $H^{-1}$ at this rate wit…
In this paper, we study exponential random graph models subject to certain constraints. We obtain some general results about the asymptotic structure of the model. We show that there exists non-trivial regions in the phase plane where the asymptotic structure is uniform and there also exists non-trivial regions in the phase plane where the asymptotic structure is non-uniform. We will get more refined results for the star model and in particular the two-star model for which a sharp transition from uniform to non-uniform structure, a stationary point and phase transitions will be obtained.
Exponential phi-mixing implies exponential psi-mixing for Markov fields on bounded degree graphs
证明有界度图上非退化k-马尔可夫随机场在指数phi混合衰减率λ>3θ时蕴含指数psi混合，并将结果应用于正则树吉布斯场（如低逆温伊辛模型）的指数psi混合推导。
Uniform Lorden-type bounds for overshoot moments for standard exponential families: small drift and an exponential correction
本文研究了标准指数族随机游走的超冲矩界，在小漂移情况下提出均匀Lorden型界，并通过指数修正改进了残余寿命界。
Strong approximations of uniform transport processes to the standard Brownian motion rely on the Skorokhod embedding of random walk with centered double exponential increments. In this note we make such an embedding explicit by means of a Poissonian scheme, which both simplifies classic constructions of strong approximations of uniform transport processes (Griego et al. (1971)) and improves their rate of strong convergence (Gorostiza et al. (1980)). We finalise by providing an extension regarding the embedding of a random walk with asymmetric double exponential increments.
On the scaling of random Tamari intervals and Schnyder woods of random triangulations (with an asymptotic D-finite trick)
本文研究大小为n的均匀随机Tamari区间，证明其上下路径顶点高度按n^{3/4}缩放并具显式极限律，经Bernardi-Bonichon双射描述了随机三角剖分Schnyder树高度，并提出基于D-finiteness的自动矩泵送技巧证明收敛性。
Uniformly continuous semigroups of sublinear transition operators
本文研究可数集上有界函数 Banach 空间中的次线性转移算子一致连续半群，证明其恰为有界次线性速率算子生成的指数族。
The exponential growth rate of non polynomially growing subgroups of $GL_d$ is conjectured to admit a uniform lower bound. This is known for non-amenable subgroups, while for amenable subgroups it is known to imply the Lehmer conjecture from number theory. In this note, we show that it is equivalent to the Lehmer conjecture. This is done by establishing a lower bound for the entropy of the random walk on the semigroup generated by the maps $x\mapsto λ\cdot x\pm 1$, where $λ$ is an algebraic number. We give a bound in terms of the Mahler measure of $λ$. We also derive a bound on the dimension o…
Uniform in time convergence of numerical schemes for stochastic differential equations via Strong Exponential stability: Euler methods, Split-Step and Tamed Schemes
本文提出一个充分且必要的准则，证明在局部Lipschitz和严格单调条件下，显式/隐式欧拉、分裂步和驯化欧拉等数值格式对SDE解的概率分布实现时间一致收敛，并通过数值算例验证。
Spatial analyticity and exponential decay of Fourier modes for the stochastic Navier-Stokes equation
该研究构造了仅需初始涡量有界的二维、三维随机Navier-Stokes方程局部时间空间实解析解，还证明了解全局存在时有限维Galerkin近似的指数衰减性，且该衰减在时间、初始涡量、噪声系数上均一致。
This article is concerned with the exponential stability and the uniform propagation of chaos properties of a class of Extended Ensemble Kalman-Bucy filters with respect to the time horizon. This class of nonlinear filters can be interpreted as the conditional expectations of nonlinear McKean Vlasov type diffusions with respect to the observation process. In contrast with more conventional Langevin nonlinear drift type processes, the mean field interaction is encapsulated in the covariance matrix of the diffusion. The main results discussed in the article are quantitative estimates of the expo…
We investigate the properties of uniform doubly stochastic random matrices, that is non-negative matrices conditioned to have their rows and columns sum to 1. The rescaled marginal distributions are shown to converge to exponential distributions and indeed even large sub-matrices of side-length $o(n^{1/2-ε})$ behave like independent exponentials. We determine the limiting empirical distribution of the singular values the the matrix. Finally the mixing time of the associated Markov chains is shown to be exactly 2 with high probability.
An Exponential Concentration Inequality for the Components of a Uniform Random Vector on the Sphere
本文证明了球面上均匀随机向量分量的经验CDF以指数速度集中于标准高斯CDF，并给出了具有显式函数和常数的有限样本统计保证。
On tightness and exponential tightness in generalised Jackson networks
本文通过统一证明，探讨了广义杰克逊网络中平稳队列长度序列的紧性与指数紧性，涵盖大偏差、正常偏差和中等偏差等多种情形。
By adopting the coupling by reflection and choosing an auxiliary function which is convex near infinity, we establish the exponential convergence of diffusion semigroups $(P_t)_{t\ge0}$ with respect to the standard $L^p$-Wasserstein distance for all $p\in[1,\infty)$. In particular, we show that for the Itô stochastic differential equation $$\d X_t=\d B_t+b(X_t)\,\d t,$$ if the drift term $b$ satisfies that for any $x,y\in\R^d$, $$\langle b(x)-b(y),x-y\rangle\le \begin{cases} K_1|x-y|^2,& |x-y|\le L; -K_2|x-y|^2,& |x-y|> L \end{cases}$$ holds with some positive constants $K_1$, $K_2$ and $L>0$,…
On uniform exponential ergodicity of Markovian multiclass many-server queues in the Halfin-Whitt regime
本文研究了在Halfin-Whitt regime中重载的马尔可夫多类多服务器队列的遍历性质，通过Lyapunov函数方法统一建立了极限扩散和预极限扩散缩放队列过程的Foster-Lyapunov方程，证明了当备用容量参数为正时，扩散过程在所有平稳马尔可夫控制下均匀指数遍历，并且不变概率测度具有均匀指数尾部特性。
Longitudinal Network Models and Permutation-Uniform Markov Chains
该研究刻画了纵向网络马尔可夫链联合分布为同参数指数族的条件，证明其置换均匀子类等价于独立同分布序列，将结论应用于时序指数随机图模型，提出新模型并简化了相关模型的分析。
Exponential rate of decay of correlations of equilibrium states associated with non-uniformly expanding circle maps
本文研究带无差别不动点的非一致扩张圆周映射，证明当连续性模Ω满足特定极限条件时，系统关于以ω为连续性模的势函数所对应的唯一平衡态具有关联指数衰减，其核心在于转移算子的谱隙性质，并应用于Manneville-Pomeau族。
We consider first passage percolation on certain isotropic random graphs in $\mathbb{R}^d$. We assume exponential concentration of passage times $T(x,y)$, on some scale $σ_r$ whenever $|y-x|$ is of order $r$, with $σ_r$ "growning like $r^χ$" for some $0<χ<1$. Heuristically this means transverse wandering of geodesics should be at most of order $Δ_r = (rσ_r)^{1/2}$. We show that in fact uniform versions of exponential concentration and wandering bounds hold: except with probability exponentially small in $t$, there are no $x,y$ in a natural cylinder of length $r$ and radius $KΔ_r$ for which eit…
We investigate the uniform reshuffling model for money exchanges: two agents picked uniformly at random redistribute their dollars between them. This stochastic dynamics is of mean-field type and eventually leads to a exponential distribution of wealth. To better understand this dynamics, we investigate its limit as the number of agents goes to infinity. We prove rigorously the so-called propagation of chaos which links the stochastic dynamics to a (limiting) nonlinear partial differential equation (PDE). This deterministic description, which is well-known in the literature, has a flavor of th…
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