Weak Convergence of Laws on R^{K} with Common Marginals
Alessio Sancetta
我们提出了一个拓扑等价的积分度量（Rachev，1991，Muller，1997），度量弱收敛的法律与共同的边缘。这个结果与应用有关，如几个简单的例子所示。
We present a result on topologically equivalent integral metrics (Rachev, 1991, Muller, 1997) that metrize weak convergence of laws with common marginals. This result is relevant for applications, as shown in a few simple examples.
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Max-semistable extreme value laws for autoregressive processes with Cantor-like marginals
本文研究边缘分布类似康托尔函数的自动回归过程，证明其边缘分布属于极大半稳定分布的吸引域，并通过引入极值指数得到该过程的极值律，同时揭示其与确定性动力系统极值的联系。
Convergence to Stable Laws in Relative Entropy
对于独立同分布的和，建立了关于相对熵的稳定律的收敛性.随机变量。
Martingale marginals do not always determine convergence
Baez-Duarte（1971）和Gilat（1972）给出了鞅依概率收敛（因此依分布收敛）但不是几乎必然收敛的例子。这里这样一个鞅构造一致有界增量，并提供了一个结构的两个鞅相同的边缘，其中一个几乎必然收敛，而另一个不收敛的概率。
In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev $ζ$-metric is used. The obtained results extend the known convergence rate estimates for geometric random sums (in the famous R{é}nyi theorem) to a considerably wider class of random indices with mixed Poisson distributions including, e. g., those with the (generalized) negative binomial distribution.
Fisher information and convergence to stable laws
研究了独立同分布和在相对Fisher信息下收敛于稳定律的问题。随机变量。
Convergence of finite-dimensional laws of the weighted quadratic variations process for some fractional Brownian sheets
本文提出并证明了一类分数布朗单的二次变差过程的有限维律的一个中心极限定理。本文的主要工具是Nourdin和Nualart基于Malliavin演算开发的方法。
On weak convergence of quasi-infinitely divisible laws
本文研究了拟无穷可分律的弱收敛准则，在自然假设下将其弱收敛与 Lévy--Khinchine 谱函数的一种特殊收敛联系起来，补充了 Lindner、Pan 和 Sato（2018）的已知结果。
Marginal Stationary Distributions and Convergence Rates of Higher Order Markov Chains
研究正则高阶马尔可夫链，解释其极限概率分布为边际平稳分布，证明其性质并建立收敛速率结果，还引入边际混合时间，扩展了一阶马尔可夫链混合时间概念到高阶链。
For a given set of random variables $X_1,\ldots,X_d$ we seek as large a family as possible of random variables $Y_1,\ldots,Y_d$ such that the marginal laws and the laws of the sums match: $Y_i\,{\buildrel d \over =}\,X_i$ and $\sum_iY_i\,{\buildrel d \over =}\,\sum_iX_i$. Under the assumption that $X_1,\ldots,X_d$ are independent and belong to any of the Meixner classes, we give a full characterisation of the random variables $Y_1,\ldots,Y_d$ and propose a practical construction by means of a finite mean square expansion. When $X_1,\ldots,X_d$ are identically distributed but not necessarily in…
Convergence to stable laws in the space $D$
研究了独立同分布的中心和与正规和的收敛性。随机元的空间$\mathcal{D}$的c{á}dl{á}g函数赋予Skorohod的$J\_1$拓扑，稳定分布在$\mathcal D$。我们的结果是基于度量空间上正则变分的概念和点过程收敛性。我们提供了一些应用，特别是更新-奖励过程的经验过程。
Limit Laws for Maxima of Contracted Stationary Gaussian Sequences
本文的主要结果是压缩平稳高斯随机序列最大值的弱极限和强极限。由于随机收缩，我们引入了一个修改的伯曼条件，这是足够的弱收敛的最大值的缩放样本。在更强的假设下，弱收敛被加强为几乎收敛。
We formulate conditions for convergence of Laws of Large Numbers and show its links with of the parts of mathematical analysis such as summation theory, convergence of orthogonal series. We present also applications of the Law of Large Numbers such as Stochastic Approximation, Density and Regression Estimation, Identification.
In this work we investigate the asymptotic behaviour of weighted partial sums of a particular class of random variables related to Oppenheim series expansions. More precisely, we verify convergence in probability as well as almost sure convergence to a strictly positive and finite constant without assuming any dependence structure or the existence of means. Results of this kind are known as exact weak and exact strong laws.
Convergence to Stable Laws and a Local Limit Theorem for Products of Positive Random Matrices
本文研究独立同分布非负矩阵乘积的范数余循环与方向的联合渐近行为，证明其向稳定律收敛，并建立给出精确收敛速率的局部极限定理。
Convergence of approximations to stochastic scalar conservation laws
我们开发了一个一般框架的近似随机标量守恒律的分析。我们的目的是证明，在最小的一致性性质和界限，这样的近似收敛到一个随机标量守恒律的解决方案。弱概率收敛模式是依律收敛，在这种情况下是最自然的。我们还使用了动力学公式和鞅方法。我们的结果被应用于文[15]中有限体积法的收敛性。
Central Limit Theorem with Exchangeable Summands and Mixtures of Stable Laws as Limits
研究了可交换随机变量赋范和的收敛律问题。首先，研究问题w.r.t.阵列的可交换的随机变量，和特殊的作用所发挥的混合物的产品的稳定的法律-作为限制在法律的规范和在不同的行的阵列-强调。然后给出了在上述测度类中收敛到特定形式的充要条件。此外，还证明了单行和收敛的充分条件。最后，一个潜在的有用的变体的制定的结果刚刚总结了简要概述，一个更完整的研究，它被推迟到未来的工作。
Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms
我们研究了伪边缘马尔可夫链蒙特卡罗算法的收敛性（Andrieu和Roberts [Ann. Statistics. 37（2009）697-725]）。我们发现，伪边际算法的渐近方差总是至少一样大的边际算法。我们表明，如果边际链承认（右）频谱间隙和权重（目标密度的归一化估计）是一致有界的，那么伪边际链有一个频谱间隙。在许多情况下，类似的结果也适用于绝对谱隙，它等价于几何遍历性。我们还考虑无界的权重分布和恢复多项式收敛速度在更具体的情况下，当边缘算法是一致遍历或独立的大都市黑斯廷斯或随机行走的大都市针对超指数密度与规则的轮廓。我们的几何收敛速度和多项式收敛速度的结果隐含中心极限定理。我们还证明了在一般条件下，如果估计的精度增加，伪边际算法的渐近方差收敛到边际算法的渐近方差。
Convergence of a stochastic particle approximation for fractional scalar conservation laws
给出了求解一类具有分数阶扩散和非线性漂移的偏微分方程的概率数值方法。这个方程的概率解释使用了一个由Lévy α稳定过程驱动的粒子系统，并通过其经验累积分布函数与它们的漂移相互作用。我们显示收敛到相应的欧拉方案的解决方案。
Convergence and regularity of probability laws by using an interpolation method
在[18]中，Fournier和Printems建立了一种方法，可以证明具有Hölder连续系数的某些随机方程的解的绝对连续性。这当然是遥不可及的使用已经经典的概率方法的基础上Malliavin演算。在文献[11]中，Debussche和Romito使用了一些Besov空间技巧，以实质性地改进Fournier和Printems的结果。在我们的论文中，我们表明，这种问题自然适合在插值空间的框架：我们证明了一个插值不等式（见命题2.5），它允许状态（甚至略有改善）上述绝对连续性的结果。此外，事实证明，上述插值不等式在一个完全不同的框架中有应用：我们使用它来估计一些收敛定理中的总方差距离误差。
Weak convergence of continuous-state branching processes with large immigration
本文研究带大尾部移民的连续状态分支过程的弱收敛，在繁殖律一阶矩有限条件下建立泛函极限定理，并识别出四类极限过程。
Speed of convergence for laws of rare events and escape rates
得到了一般弱相依随机过程收敛于极值律的收敛速率的误差项。误差项对“时间”和“长度”尺度的依赖性是非常明确的。专门的数据来自一类动力系统，我们发现更详细的误差项，其中一个应用是考虑逃逸率通过小孔在这些系统。
This paper deals with the rate of convergence in 1-Wasserstein distance of the marginal law of a Brownian motion with drift conditioned not to have reached 0 towards the Yaglom limit of the process. In particular it is shown that, for a wide class of initial measures including Dirac measures and probability measures with compact support, the Wasserstein distance decays asymptotically as 1/t. Likewise, this speed of convergence is recovered for the convergence of marginal laws conditioned not to be absorbed up to a horizon time towards the Bessel-$3$ process, when the horizon time tends to infi…
Convergence to stable laws for multidimensional stochastic recursions: the case of regular matrices
给定i.i.d.的序列$(M_{n},Q_{n})_{n\ge 1}$，本文考虑随机差分方程$(M,Q) \in GL(d, \R) \times \R^d$，并假设$κ>0$的存在性使得$$ \lim_{n \to \infty}(\E{\norm{M_1 ... M_n}^κ})^{\frac{1}{n}} = 1 .$$.在适当的假设下，证明了序列$S_n = R_1 + ... + R_n$经过适当的规范化后，依律收敛于指标为$κ$的多维稳定分布.作为一个副产品，我们证明了唯一的固定解决方案$R$的RDE是定期变化的指数$κ$，并给出了一个精确的描述其尾部措施。这扩展了先前的工作http://arxiv.org/abs/1009.1728v3。
On convergence to non normal laws for ergodic random fields
研究遍历随机场中鞅差部分和的极限分布非正态性，将正态性的充要条件从二维推广到任意有限维，并给出改进的中心极限定理。
Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps
我们考虑由随机递归$X_{n}^{x}=\p_{θ_{n}}(X_{n-1}^{x})$定义的$\R^d$上的马尔可夫链$\{X_n^x\}_{n=0}^\infty$，从$x\in\R^d$开始，其中$θ_{1}, θ_{2},...$是独立同分布的。取值于度量空间$(Θ, \mathfrak{r}),$和$\p_{θ_{n}}:\R^d\mapsto\R^d$的随机变量是Lipschitz映射。假设马尔可夫链有唯一的平稳测度PDMATHAMT。在对$\p_{θ_n}$的适当假设下，我们将证明测度$ν$具有指数为$α>0$的重尾，即$ν(\{x\in\R^d: |x|>t\})\asymp t^{-α}$。利用这一结果，我们表明，适当规范的Birkhoff和$S_n^x=\sum_{k=1}^n X_k^x$，收敛于法律的$α$ -稳定的法律$α\in(0, 2]$。
Convergence Rate of the Solution of Multi-marginal Schrodinger Bridge Problem with Marginal Constraints from SDEs
本文研究了边际约束源自漂移项随时间变化的SDE的多边际薛定谔桥问题，证明了其解在KL散度意义下以$O(m^{-1})$速率收敛到SDE解的分布，推广了已有研究。
An Autoregressive Model with Semi-stable Marginals
半稳定律族是半自分解的。因此，他们有资格模型平稳的一阶自回归计划。这些自回归计划与半稳定的边缘和半自相似过程之间的联系。
Weak convergence from projected laws on a positive-measure set of directions
研究通过正测度方向上的投影收敛推导概率测度的弱收敛，证明在表面几乎每个方向上投影定律矩确定时，正测度方向上的投影收敛即可保证全局弱收敛。
Let $\{X_n\}_n$ be a sequence of freely independent, identically distributed non-commutative random variables. Consider a sequence $\{W_n\}_n$ of the renormalized spectral maximum of random variables $X_1,\cdots, X_n$. It is known that the renormalized spectral maximum $W_n$ converges to the free extreme value distribution under certain conditions on the distribution function. In this paper, we provide a rate of convergence in the Kolmogorov distance between a distribution function of $W_n$ and the free extreme value distribution.
In this paper, we give estimates for the speed of convergence towards a limiting stable law in the recently introduced setting of mod-$ϕ$ convergence. Namely, we define a notion of zone of control, closely related to mod-$ϕ$ convergence, and we prove estimates of Berry-Esseen type under this hypothesis. Applications include: the winding number of a planar Brownian motion; classical approximations of stable laws by compound Poisson laws; examples stemming from determinantal point processes (characteristic polynomials of random matrices and zeroes of random analytic functions); sums of variables…
On convergence of entropy of distribution functions in the max domain of attraction of max stable laws
最大稳定律是独立同分布随机变量（RVS）的线性归一化部分最大值的极限律。这些类似于稳定的法律，这是限制法律规范化部分和的iid rvs。本文研究了最大稳定律的最大吸引区域中分布函数在线性正规化下的熵极限定理。更具体地说，我们研究的问题的香农熵的线性归一化的部分最大值的iid rvs的收敛到相应的极限熵时，线性归一化的部分最大值收敛到一些非退化rv。我们能够证明，香农熵不仅收敛，但事实上，在某些情况下，增加到极限熵。我们讨论几个例子。我们还研究了第k个上限的类似结果。
Space-time transport of Brownian exit laws
本文系统研究p-布朗收敛，将其纳入最优输运理论框架，并得到位似域情形下的渐近行为。
In this paper, we are concerned with a operator splitting scheme for linear fractional and fractional degenerate stochastic conservation laws driven by multiplicative Levy noise. More specifically, using a variant of classical Kruzkov's doubling of variable approach, we show that the approximate solutions generated by the splitting scheme converges to the unique stochastic entropy solution of the underlying problems.Finally, the convergence analysis is illustrated by several numerical examples.
Convergence of the Finite Volume Method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation
本文研究了在乘性噪声下的标量守恒律的有限体积法收敛性，采用动力学公式方法证明了显式时间有限体积法的收敛性。
In the framework of classical probability, we consider the normal product distribution $F_\infty \sim N_1 \times N_2$ where $N_1, N_2$ are two independent standard normal random variable, and in the setting of free probability, $F_\infty \sim \left( S_1 S_2 + S_2 S_1 \right)/\sqrt{2}$ known as {\it tetilla law} \cite{d-n}, where $S_1, S_2$ are freely independent normalized semicircular random variables. We provide novel characterization of $F_\infty$ within the second Wiener (Wigner) chaos. More precisely, we show that for any generic element $F$ in the second Wiener (Wigner) chaos with varian…
We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and Hölder continuity of the optimal coupling in $p$-Wasserstein distance are obtained under general conditions including quadratic costs and unbounded marginals. The results for the value extend to regularization by an arbitrary divergence. As an application, we show convergence of Sinkhorn's algorithm in Wasserstein sense, including for quadratic cost. Two techniques are presented: The first compares an optimal coupling with its so-called shadow, a coupling in…
Exponential convergence of Sinkhorn algorithm for entropy martingale optimal transport
本文证明在紧支撑、严格凸序等条件下，熵鞅最优传输的Sinkhorn算法在相对熵意义下指数收敛，并给出优化器的存在唯一性及稳定性估计。
We investigate yet another approach to understand the limit behaviour of Brownian motion conditioned to stay within a tubular neighbourhood around a closed and connected submanifold of a Riemannian manifold. In this context, we identify a second order generator subject to Dirichlet conditions on the boundary of the tube and study its associated semigroups. After a suitable rescaling and renormalization procedure, we obtain convergence of these semigroups, both in $L^2$ and in Sobolev spaces of arbitrarily large index, to a limit semigroup, as the tube diameter tends to zero. As a byproduct, we…
Central Limit Theorem and convergence to stable laws in Mallows distance
在Malows（$L^r$ -Wasserstein）距离下给出了经典中心极限定理的一个新证明.我们的证明是基本的，在这个意义上，它不需要复分析，而是利用了一个简单的次加性不等式与此度量。关键是要分析平等的情况。我们提供了一些结果的收敛速度。我们还考虑了收敛到稳定分布，并得到了这样的收敛速度的一个界。
We study the stability of entropically regularized optimal transport with respect to the marginals. Given marginals converging weakly, we establish a strong convergence for the Schrödinger potentials describing the density of the optimal couplings. When the marginals converge in total variation, the optimal couplings also converge in total variation. This is applied to show that Sinkhorn's algorithm converges in total variation when costs are quadratic and marginals are subgaussian, or more generally for all continuous costs satisfying an integrability condition.
Logical convergence laws via stochastic approximation and Markov processes
本文证明有界度数的均匀连接随机图服从一阶收敛律，通过马尔可夫链刻画一阶等价类动力学，并借助随机逼近过程证明其极限分布存在。
Convergence to Stable Laws for Products of Random Matrices
本文研究了随机矩阵乘积的log-norm满足广义中心极限定理，并通过构造关键时间点证明了矩增益，同时探讨了高阶扩展的可能。
We present a novel analogue for finite exchangeable sequences of the de Finetti, Hewitt and Savage theorem and investigate its implications for multi-marginal optimal transport (MMOT) and Bayesian statistics. If $(Z_1,...,Z_N)$ is a finitely exchangeable sequence of $N$ random variables taking values in some Polish space $X$, we show that the law $μ_k$ of the first $k$ components has a representation of the form $μ_k=\int_{{\mathcal P}_{\frac{1}{N}}(X)} F_{N,k}(λ) \, \mbox{d} α(λ)$ for some probability measure $α$ on the set of $1/N$-quantized probability measures on $X$ and certain universal …
A more general notion of weight called admissible is introduced and then an investigation is carried out on the a.e. convergence of weighted strong laws of large numbers and their applications to weighted one-sided ergodic Hilbert transforms on vector-valued $L_p$-spaces. Furthermore, the obtained results are applied to the a.e. convergence of random modulated weighted ergodic series, which is also new in the classical setting.
This paper exploit the equivalence between the Schrödinger Bridge problem and the entropy penalized optimal transport in order to find a different approach to the duality, in the spirit of optimal transport. This approach results in a priori estimates which are consistent in the limit when the regularization parameter goes to zero. In particular, we find a new proof of the existence of maximizing entropic-potentials and therefore, the existence of a solution of the Schrödinger system. Our method extends also when we have more than two marginals: we can provide an alternative proof of the conve…
Wall laws for viscous flows in 3D randomly rough pipes: optimal convergence rates and stochastic integrability
本文研究了三维随机粗糙管道中粘性层流的壁面律与有效近似，将二维通道流的结果推广至John型随机边界三维管道流，建立了最优收敛速率与改进的随机可积性，并证明了精化版的Poiseuille定律。
Markovian projections for Itô semimartingales with jumps
本文为带跳 Itô 半鞅构造了匹配一维边际分布的 Markovian 投影，证明其边际分布流为非局部 FPKEs 的解，并展示了其在构建兼具局部与随机特征的校准扩散/跳跃模型中的应用。
Convergence Rates for Regularized Optimal Transport via Quantization
本文研究散度正则化最优传输在正则化参数趋于零时的收敛速率，提出基于量化与鞅耦合的新方法，适用于非紧致边际，并获得了熵正则化2-Wasserstein距离的尖锐首阶项。
We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size $N$ converge to the Tracy--Widom laws at a rate $O(N^{-1/3+ω})$, as $N$ tends to infinity. For Wigner matrices this improves the previous rate $O(N^{-2/9+ω})$ obtained by Bourgade [5] for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős, Yau and Yin [19] to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow indu…
Quantitative contraction rates for Sinkhorn's algorithm: beyond bounded costs and compact marginals
本文研究了Sinkhorn迭代在实数空间上的指数收敛性，无需假设有界成本或紧边际，首次在一般情况下证明了Sinkhorn算法的指数收敛性。
We establish a quantitative version of the Tracy--Widom law for the largest eigenvalue of high dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix $X^*X$ converge to its Tracy--Widom limit at a rate nearly $N^{-1/3}$, where $X$ is an $M \times N$ random matrix whose entries are independent real or complex random variables, assuming that both $M$ and $N$ tend to infinity at a constant rate. This result improves the previous estimate $N^{-2/9}$ obtained by Wang [73]. Our proof relies on a Green function comp…
Convergence of stochastic integrals with applications to transport equations and conservation laws with noise
针对时间变量弱收敛下随机积分极限的收敛难题，提出基于均匀均值 $L^1$ 时间平移估计的方法，并证明了其在随机输运方程和守恒律中的稳定性。
$q_x$-Bass Martingales: Optimality, Existence and Convergence
本研究扩展$q$-Bass鞅为$q_x$-Bass鞅，通过弱最优传输问题定义其为转移核最接近参考核族的鞅耦合，建立强对偶后提出计算算法并证其收敛与存在性，一维情形下简化为不动点迭代。
We study the central limit theorem in the non-normal domain of attraction to symmetric $α$-stable laws for $0<α\leq2$. We show that for i.i.d. random variables $X_i$, the convergence rate in $L^\infty$ of both the densities and distributions of $\sum_i^n X_i/(n^{1/α}L(n))$ is at best logarithmic if $L$ is a non-trivial slowly varying function. Asymptotic laws for several physical processes have been derived using central limit theorems with $\sqrt{n\log n}$ scaling and Gaussian limiting distributions. Our result implies that such asymptotic laws are accurate only for exponentially large $n$.
In this paper we prove the weak convergence, in a high-temperature phase, of the finite marginals of the Gibbs measure associated to a symmetric spherical spin glass model with correlated couplings towards an explicit asymptotic decoupled measure. We also provide upper bounds for the rate of convergence in terms of the one of the energy per variable. Furthermore, we establish a concentration inequality for bounded functions under a higher temperature condition. These results are exemplified by analysing the asymptotic behaviour of the empirical mean of coordinate-wise functions of samples from…
A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm
该研究提出基于半凹性的方法，分析熵最优传输计划的稳定性与Sinkhorn算法的指数收敛性，给出无需代价有界的收敛充分条件，在多种分布和代价场景下得到新的紧最优收敛速率。
Convex geometry of diffusion laws and gradient flows for stochastic control
本文研究弱随机控制的梯度型动力学，通过路径律的凸几何构造镜像流与自然梯度流，证明其收敛性、稳定性及有限粒子近似的指数收敛和O(1/N)误差。
On the Convergence Rate of Sinkhorn's Algorithm
本文研究熵正则化最优传输问题的 Sinkhorn 算法，证明其相对熵、对偶次优性和边缘熵的收敛速率分别为 $O(t^{-1})$、$O(t^{-1})$ 和 $O(t^{-2})$，导出不随正则化参数指数恶化的非渐近界，并给出最优耦合的稳定性结果。
Stochastic Local Intensity Loss Models with Interacting Particle Systems
从Schönbucher（2005）的工作中可以知道，损失过程的边际律可以被一个单位时间递增的非齐次马尔可夫过程匹配，其确定性跳跃强度称为局部强度。随机局部强度（SLI）模型，如Arnsdorf和Halperin（2008）提出的模型，允许在保持相同的边际律的同时获得随机跳跃强度。这些模型涉及一个非线性的跳跃。本文的第一个贡献是证明了这类过程的存在性和唯一性。这是通过一个相互作用的粒子系统，其收敛速度向非线性收敛进行了分析。其次，这种方法提供了一个强大的方法来计算路径的期望与SLI模型：我们表明，计算成本大致相同的原油蒙特卡洛算法的标准SDES。
Semi-discretization for stochastic scalar conservation laws with multiple rough fluxes
本文提出半离散化方法用于求解具有多粗糙时间依赖的非均匀通量的随机标量守恒律，基于Brenier运输-坍缩算法和粗糙路径定义的特征线，证明了强L^1收敛性及收敛速度。
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