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From p-Wasserstein bounds to moderate deviations 从p-Wasserstein界到中等偏差
IF 1.4 3区 数学 Q2 STATISTICS & PROBABILITY Pub Date : 2022-05-26 DOI: 10.1214/23-ejp976
Xiao Fang, Yuta Koike
We use a new method via $p$-Wasserstein bounds to prove Cram'er-type moderate deviations in (multivariate) normal approximations. In the classical setting that $W$ is a standardized sum of $n$ independent and identically distributed (i.i.d.) random variables with sub-exponential tails, our method recovers the optimal range of $0leq x=o(n^{1/6})$ and the near optimal error rate $O(1)(1+x)(log n+x^2)/sqrt{n}$ for $P(W>x)/(1-Phi(x))to 1$, where $Phi$ is the standard normal distribution function. Our method also works for dependent random variables (vectors) and we give applications to the combinatorial central limit theorem, Wiener chaos, homogeneous sums and local dependence. The key step of our method is to show that the $p$-Wasserstein distance between the distribution of the random variable (vector) of interest and a normal distribution grows like $O(p^alpha Delta)$, $1leq pleq p_0$, for some constants $alpha, Delta$ and $p_0$. In the above i.i.d. setting, $alpha=1, Delta=1/sqrt{n}, p_0=n^{1/3}$. For this purpose, we obtain general $p$-Wasserstein bounds in (multivariate) normal approximations using Stein's method.
我们使用一种新的方法,通过$p$-Waserstein界来证明(多元)正态近似中的Cram’er型中偏差。在$W$是具有次指数尾的$n$独立同分布(i.i.d.)随机变量的标准和的经典设置中,我们的方法恢复了$0leq x=o(n^{1/6})$的最优范围和$P(W>x)/(1-Phi(x))到1$的近似最优错误率$o(1)(1+x)(log n+x^2)/sqrt{n}$,其中$Phi$是标准正态分布函数。我们的方法也适用于因随机变量(向量),并应用于组合中心极限定理、维纳混沌、齐次和和局部依赖。我们方法的关键步骤是证明,对于一些常数$alpha、Delta$和$p_0$,感兴趣的随机变量(向量)的分布和正态分布之间的$p$-Waserstein距离增长为$O(p^alphaDelta)$、$1leq pleq p_0$。在上述i.i.d.设置中,$alpha=1,Delta=1/sqrt{n},p_0=n^{1/3}$。为此,我们使用Stein方法获得了(多元)正态近似中的一般$p$-Waserstein界。
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引用次数: 5
Spatial populations with seed-banks in random environment: III. Convergence towards mono-type equilibrium 随机环境中具有种子库的空间种群:III.向单型平衡的收敛
IF 1.4 3区 数学 Q2 STATISTICS & PROBABILITY Pub Date : 2022-05-24 DOI: 10.1214/23-EJP922
S. Nandan
We consider the spatially inhomogeneous Moran model with seed-banks introduced in den Hollander and Nandan (2021). Populations comprising $active$ and $dormant$ individuals are structured in colonies labelled by $mathbb{Z}^d,~dgeq 1$. The population sizes are drawn from an ergodic, translation-invariant, uniformly elliptic field that form a random environment. Individuals carry one of two types: $heartsuit$, $spadesuit$. Dormant individual resides in what is called a seed-bank. Active individuals exchange type from seed-bank of their own colony and resample type by choosing parent from the active populations according to a symmetric migration kernel. In den Hollander and Nandan (2021) by using a dual (an interacting coalescing particle system), we showed that the spatial system exhibits a dichotomy between $clustering$ (mono-type equilibrium) and $coexistence$ (multi-type equilibrium). In this paper we identify the domain of attraction for each mono-type equilibrium in the clustering regime for a $fixed$ environment. We also show that when the migration kernel is $recurrent$, for a.e. realization of the environment, the system with an initially $consistent$ type distribution converges weakly to a mono-type equilibrium in which the fixation probability to type-$heartsuit$ configuration does not depend on the environment. A formula for the fixation probability is given in terms of an annealed average of type-$heartsuit$ densities in dormant and active population biased by ratio of the two population sizes at the target colony. For the proofs, we use duality and environment seen by particle introduced in Dolgopyat and Goldsheid (2019) for RWRE on a strip. A spectral analysis of Markov operator yields quenched weak convergence of the environment process associated with single-particle dual to a reversible ergodic distribution which we transfer to the spatial system by using duality.
我们考虑了den Hollander和Nandan(2021)中引入的具有种子库的空间非均匀Moran模型。由$active$和$sleeved$个体组成的群体在用$mathbb{Z}^d,~dgeq 1$标记的群体中构造。种群大小是从遍历的、平移不变的、形成随机环境的一致椭圆场中得出的。个人携带两种类型之一:$heartsuit$、$spadesuit$。休眠个体居住在所谓的种子库中。活跃个体根据对称迁移核从自己群体的种子库中交换类型,并通过从活跃群体中选择亲本来重新采样类型。在den Hollander和Nandan(2021)中,通过使用对偶(一种相互作用的聚结粒子系统),我们表明空间系统表现出$clustering$(单型平衡)和$共存$(多型平衡)之间的二分法。在本文中,我们确定了$fixed$环境的聚类机制中每个单型平衡的吸引域。我们还表明,当迁移内核是$recurrent$时,例如,对于环境的实现,具有初始$consistent$类型分布的系统弱收敛于单类型平衡,在该平衡中,对类型-$heartsuit$配置的固定概率不依赖于环境。根据休眠种群和活动种群中-$heartsuit$型密度的退火平均值,给出了固定概率的公式,该平均值受目标群体两个种群大小之比的偏差。对于证明,我们使用Dolgopyat和Goldsheid(2019)中引入的粒子在条带上的RWRE所看到的对偶性和环境。马尔可夫算子的谱分析产生了与单粒子对偶相关的环境过程到可逆遍历分布的淬灭弱收敛性,我们使用对偶将其转移到空间系统。
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引用次数: 0
Scaling limit for a second-order particle system with local annihilation 具有局部湮灭的二阶粒子系统的标度极限
IF 1.4 3区 数学 Q2 STATISTICS & PROBABILITY Pub Date : 2022-05-15 DOI: 10.1214/23-ejp973
Ruojun Huang
For a second-order particle system in $mathbb R^d$ subject to locally-in-space pairwise annihilation, we prove a scaling limit for its empirical measure on position and velocity towards a degenerate elliptic partial differential equation. Crucial ingredients are Green's function estimates for the associated hypoelliptic operator and an It^o-Tanaka trick.
对于$mathbb R^d$中的二阶粒子系统,在空间中局部成对湮灭,我们向退化椭圆偏微分方程证明了其位置和速度经验测度的标度极限。关键成分是相关的次椭圆算子的格林函数估计和It-o-Tanaka技巧。
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引用次数: 1
Creeping of Lévy processes through curves lsamvy的爬行过程通过曲线
IF 1.4 3区 数学 Q2 STATISTICS & PROBABILITY Pub Date : 2022-05-13 DOI: 10.1214/23-ejp942
L. Chaumont, Thomas Pellas
A L'evy process is said to creep through a curve if, at its first passage time across this curve, the process reaches it with positive probability. We first study this property for bivariate subordinators. Given the graph ${(t,f(t)):tge0}$ of any continuous, non increasing function $f$ such that $f(0)>0$, we give an expression of the probability that a bivariate subordinator $(Y,Z)$ issued from 0 creeps through this graph in terms of its renewal function and the drifts of the components $Y$ and $Z$. We apply this result to the creeping probability of any real L'evy process through the graph of any continuous, non increasing function at a time where the process also reaches its past supremum. This probability involves the density of the renewal function of the bivariate upward ladder process as well as its drift coefficients. We also investigate the case of L'evy processes conditioned to stay positive creeping at their last passage time below the graph of a function. Then we provide some examples and we give an application to the probability of creeping through fixed levels by stable Ornstein-Uhlenbeck processes. We also raise a couple of open questions along the text.
如果在其第一次通过曲线时,过程以正概率到达曲线,则称每一个过程缓慢地通过曲线。我们首先研究了二元从属变量的这一性质。给定任意连续非递增函数$f$的图${(t,f(t)):tge0}$,使得$f(0)> $,我们给出了从0发出的二元次元$(Y,Z)$的更新函数和分量$Y$和$Z$的漂移的概率表达式。我们将这个结果应用于任何实L' every过程的爬行概率,通过任何连续的,不增加的函数的图,在该过程也达到其过去的上限时。这个概率涉及二元向上阶梯过程的更新函数的密度以及它的漂移系数。我们还研究了在函数图下的L' evs过程在其最后通过时间条件下保持正爬行的情况。然后给出了一些例子,并给出了稳定的Ornstein-Uhlenbeck过程爬过固定水平的概率的应用。我们还提出了几个悬而未决的问题。
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引用次数: 2
The SLE loop via conformal welding of quantum disks 量子盘保角焊接的SLE环
IF 1.4 3区 数学 Q2 STATISTICS & PROBABILITY Pub Date : 2022-05-10 DOI: 10.1214/23-ejp914
M. Ang, N. Holden, Xin Sun
We prove that the SLE$_kappa$ loop measure arises naturally from the conformal welding of two $gamma$-Liouville quantum gravity (LQG) disks for $gamma^2 = kappa in (0,4)$. The proof relies on our companion work on conformal welding of LQG disks and uses as an essential tool the concept of uniform embedding of LQG surfaces. Combining our result with work of Gwynne and Miller, we get that random quadrangulations decorated by a self-avoiding polygon converge in the scaling limit to the LQG sphere decorated by the SLE$_{8/3}$ loop. Our result is also a key input to recent work of the first and third coauthors on the integrability of the conformal loop ensemble.
我们证明了SLE$_kappa$环测度是由两个$gamma$-Liouville量子重力(LQG)圆盘对$gamma ^2=kappain(0.4)$的保角焊接自然产生的。该证明依赖于我们关于LQG圆盘保形焊接的配套工作,并使用LQG表面均匀嵌入的概念作为基本工具。将我们的结果与Gwynne和Miller的工作相结合,我们得到了由自回避多边形装饰的随机四边形在由SLE$_{8/3}$环装饰的LQG球面的标度极限上收敛。我们的结果也是第一和第三合著者最近关于共形环系综可积性的工作的关键输入。
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引用次数: 4
SDEs with no strong solution arising from a problem of stochastic control 随机控制问题产生的无强解SDE
IF 1.4 3区 数学 Q2 STATISTICS & PROBABILITY Pub Date : 2022-05-05 DOI: 10.1214/23-ejp995
A. Cox, Benjamin A. Robinson
We study a two-dimensional stochastic differential equation that has a unique weak solution but no strong solution. We show that this SDE shares notable properties with Tsirelson's example of a one-dimensional SDE with no strong solution. In contrast to Tsirelson's equation, which has a non-Markovian drift, we consider a strong Markov martingale with Markovian diffusion coefficient. We show that there is no strong solution of the SDE and that the natural filtration of the weak solution is generated by a Brownian motion. We also discuss an application of our results to a stochastic control problem for martingales with fixed quadratic variation in a radially symmetric environment.
我们研究了一个二维随机微分方程,它有一个唯一的弱解,但没有强解。我们证明了这个SDE与Tsirelson的没有强解的一维SDE的例子具有显著的性质。与具有非马尔可夫漂移的Tsirelson方程相反,我们考虑了具有马尔可夫扩散系数的强马尔可夫鞅。我们证明了SDE不存在强解,弱解的自然过滤是由布朗运动产生的。我们还讨论了我们的结果在径向对称环境中具有固定二次变分的鞅的随机控制问题上的应用。
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引用次数: 3
Subexponentialiy of densities of infinitely divisible distributions 无限可分分布密度的次指数性
IF 1.4 3区 数学 Q2 STATISTICS & PROBABILITY Pub Date : 2022-05-04 DOI: 10.1214/23-ejp928
Muneya Matsui
We show the equivalence of three properties for an infinitely divisible distribution: the subexponentiality of the density, the subexponentiality of the density of its L'evy measure and the tail equivalence between the density and its L'evy measure density, under monotonic-type assumptions on the L'evy measure density. The key assumption is that tail of the L'evy measure density is asymptotic to a non-increasing function or is eventually non-increasing. Our conditions are novel and cover a rather wide class of infinitely divisible distributions. Several significant properties for analyzing the subexponentiality of densities have been derived such as closure properties of [ convolution, convolution roots and asymptotic equivalence ] and the factorization property. Moreover, we illustrate that the results are applicable for developing the statistical inference of subexponential infinitely divisible distributions which are absolutely continuous.
在L维测度密度的单调类型假设下,我们证明了无穷可分分布的三个性质的等价性:密度的次指数性、其L维测度的密度的次幂指数性以及密度与其L维测度之间的尾等价性。关键的假设是L’evy测度密度的尾部是非递增函数的渐近性,或者最终是非递增的。我们的条件是新颖的,涵盖了一类相当广泛的无限可分分布。导出了分析密度次指数性的几个重要性质,如[卷积,卷积根和渐近等价]的闭包性质和因子分解性质。此外,我们还证明了这些结果适用于发展绝对连续的亚指数无限可分分布的统计推断。
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引用次数: 2
A diploid population model for copy number variation of genetic elements 遗传元件拷贝数变异的二倍体群体模型
IF 1.4 3区 数学 Q2 STATISTICS & PROBABILITY Pub Date : 2022-04-23 DOI: 10.1214/23-ejp934
P. Pfaffelhuber, A. Wakolbinger
We study the following model for a diploid population of constant size $N$: Every individual carries a random number of (genetic) elements. Upon a reproduction event each of the two parents passes each element independently with probability $tfrac 12$ on to the offspring. We study the process $X^N = (X^N(1), X^N(2),...)$, where $X_t^N(k)$ is the frequency of individuals at time $t$ that carry $k$ elements, and prove convergence (in some weak sense) of $X^N$ jointly with its empirical first moment $Z^N$ to the ``slow-fast'' system $(Z,X)$, where $X_t = text{Poi}(Z_t)$ and $Z$ evolves according to a critical Feller branching process. We discuss heuristics explaining this finding and some extensions and limitations.
我们研究了以下恒定大小$N$的二倍体种群的模型:每个个体都携带随机数目的(遗传)元素。在繁殖事件中,两个亲本中的每一个都以概率$trac 12$独立地将每个元素传递给后代。我们研究了过程$X^N=(X^N(1),X^N(2),…)$,其中$X_t^N(k)$是在时间$t$携带$k$元素的个体的频率,并证明$X^N$与其经验第一矩$Z^N$联合收敛于“慢-快”系统$(Z,X)$,其中$X_t=text{Poi}(Z_t)$和$Z$根据关键的Feller分支过程演化。我们讨论了解释这一发现的启发式方法以及一些扩展和限制。
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引用次数: 0
Chaos for rescaled measures on Kac’s sphere Kac球面上重尺度测度的混沌
IF 1.4 3区 数学 Q2 STATISTICS & PROBABILITY Pub Date : 2022-04-11 DOI: 10.1214/23-ejp967
R. Cortez, H. Tossounian
In this article we study a relatively novel way of constructing chaotic sequences of probability measures supported on Kac's sphere, which are obtained as the law of a vector of $N$ i.i.d. variables after it is rescaled to have unit average energy. We show that, as $N$ increases, this sequence is chaotic in the sense of Kac, with respect to the Wasserstein distance, in $L^1$, in the entropic sense, and in the Fisher information sense. For many of these results, we provide explicit rates of polynomial order in $N$. In the process, we improve a quantitative entropic chaos result of Haurey and Mischler by relaxing the finite moment requirement on the densities from order $6$ to $4+epsilon$.
本文研究了一种较为新颖的构造Kac球上支持的概率测度混沌序列的方法,该混沌序列是由$N$ i. id个变量的向量在重新标度为具有单位平均能量后的定律得到的。我们证明,随着N的增加,这个序列在Kac意义上是混沌的,关于Wasserstein距离,在L^1意义上,在熵意义上,在Fisher信息意义上。对于其中的许多结果,我们给出了N$中多项式阶的显式速率。在此过程中,我们通过将密度的有限矩要求从$6$放宽到$4+epsilon$,改进了Haurey和Mischler的定量熵混沌结果。
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引用次数: 0
Random cubic planar graphs converge to the Brownian sphere 随机三次平面图收敛于布朗球
IF 1.4 3区 数学 Q2 STATISTICS & PROBABILITY Pub Date : 2022-03-31 DOI: 10.1214/23-EJP912
M. Albenque, 'Eric Fusy, Thomas Leh'ericy
In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere. The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances. Then, Whitney's theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit is the Brownian sphere. Curien and Le Gall have recently developed a framework to study the modification of distances in general triangulations and in their dual. By extending this framework to simple triangulations, it is shown that 3-connected cubic planar graphs with modified distances converge jointly with their dual triangulation to the Brownian sphere.
本文证明了随机连通三次平面图(分别为多图)的标度极限是布朗球。证明主要包括两个步骤。首先,由于已知的三次平面图分解为它们的3连通分量,随机三次平面图的度量结构被证明是由其唯一的线性大小的3连通分量很好地近似,并具有修改的距离。然后,惠特尼定理保证了一个3连通的三次平面图形是一个简单三角剖分的对偶,对于这个简单三角剖分,已知其尺度极限是布朗球。curen和Le Gall最近开发了一个框架来研究一般三角测量及其对偶中的距离修改。将这一框架推广到简单的三角剖分上,证明了具有修正距离的3连通三次平面图与它们的对偶三角剖分一起收敛于布朗球。
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引用次数: 3
期刊
Electronic Journal of Probability
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