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On the affine recursion on R_d+ in the critical case 临界情况下R_d+上的仿射递归
IF 0.7 4区 数学 Q4 STATISTICS & PROBABILITY Pub Date : 2021-01-01 DOI: 10.30757/ALEA.V18-37
S. Brofferio, M. Peigné, Thi da Cam Pham
We fix d ≥ 2 and denote S the semi-group of d× d matrices with non negative entries. We consider a sequence (An, Bn)n≥1 of i. i. d. random variables with values in S × R+ and study the asymptotic behavior of the Markov chain (Xn)n≥0 on R+ defined by: ∀n ≥ 0, Xn+1 = An+1Xn +Bn+1, where X0 is a fixed random variable. We assume that the Lyapunov exponent of the matrices An equals 0 and prove, under quite general hypotheses, that there exists a unique (infinite) Radon measure λ on (R+)d which is invariant for the chain (Xn)n≥0. The existence of λ relies on a recent work by T.D.C. Pham about fluctuations of the norm of product of random matrices [16]. Its unicity is a consequence of a general property, called “local contractivity”, highlighted about 20 years ago by M. Babillot, Ph. Bougerol et L. Elie in the case of the one dimensional affine recursion [1] .
我们定d≥2,并将S记为具有非负项的d× d矩阵的半群。我们考虑一个序列(An, Bn)n≥1,包含i. i. d个值在S × R+中的随机变量,并研究马尔可夫链(Xn)n≥0在R+上的渐近性:∀n≥0,Xn+1 = An+1Xn +Bn+1,其中X0是一个固定的随机变量。我们假设矩阵An的Lyapunov指数等于0,并在相当一般的假设下证明在(R+)d上存在一个唯一的(无限的)Radon测度λ,该测度对于链(Xn)n≥0是不变的。λ的存在依赖于T.D.C. Pham最近关于随机矩阵乘积范数涨落的研究[16]。它的唯一性是一个一般性质的结果,称为“局部收缩性”,大约20年前由M. babilllot, Ph. Bougerol和L. Elie在一维仿射递推的情况下强调[1]。
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引用次数: 2
On the Passage Time Geometry of theLast Passage Percolation Problem 关于最后通道渗流问题的通过时间几何
IF 0.7 4区 数学 Q4 STATISTICS & PROBABILITY Pub Date : 2021-01-01 DOI: 10.30757/alea.v18-10
Tom Alberts, E. Cator
We analyze the geometrical structure of the passage times in the last passage percolation model. Viewing the passage time as a piecewise linear function of the weights we determine the domains of the various pieces, which are the subsets of the weight space that make a given path the longest one. We focus on the case when all weights are assumed to be positive, and as a result each domain is a pointed polyhedral cone. We determine the extreme rays, facets, and two-dimensional faces of each cone, and also review a well-known simplicial decomposition of the maximal cones via the so-called order cone. All geometric properties are derived using arguments phrased in terms of the last passage model itself. Our motivation is to understand path probabilities of the extremal corner paths on rectangles in Z, but all of our arguments apply to general, finite partially ordered sets.
我们分析了最后一种通道渗流模型中通道时间的几何结构。将通过时间视为权重的分段线性函数,我们确定了各个块的域,这些块是权重空间的子集,使给定路径成为最长的路径。我们关注的是假设所有的权值都是正的,因此每个域都是一个尖多面体锥的情况。我们确定了每个锥体的极端射线、切面和二维面,并通过所谓的有序锥体回顾了一个众所周知的最大锥体的简单分解。所有几何属性都是使用根据最后通道模型本身措辞的参数派生的。我们的动机是理解Z上矩形的极值角路径的路径概率,但我们所有的论点都适用于一般的有限部分有序集合。
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引用次数: 1
Mixing properties of integer-valued GARCH processes 整数值GARCH过程的混合特性
IF 0.7 4区 数学 Q4 STATISTICS & PROBABILITY Pub Date : 2021-01-01 DOI: 10.30757/ALEA.V18-18
P. Doukhan, N. M. Khan, Michael H. Neumann
We consider models for count variables with a GARCH-type structure. Such a process consists of an integer-valued component and a volatility process. Using arguments for contractive Markov chains we prove that this bivariate process has a unique stationary regime. Furthermore, we show absolute regularity (β-mixing) with geometrically decaying coefficients for the count process. These probabilistic results are complemented by a statistical analysis and a few simulations.
我们考虑具有garch类型结构的计数变量模型。该过程由整数分量和波动过程组成。利用压缩马尔可夫链的参数,证明了该二元过程具有唯一的平稳区。此外,我们在计数过程中显示了具有几何衰减系数的绝对规律性(β-混合)。这些概率结果由统计分析和一些模拟加以补充。
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引用次数: 7
Quantization and martingale couplings 量子化与鞅耦合
IF 0.7 4区 数学 Q4 STATISTICS & PROBABILITY Pub Date : 2020-12-18 DOI: 10.30757/alea.v19-01
B. Jourdain, G. Pagès
Quantization provides a very natural way to preserve the convex order when approximating two ordered probability measures by two finitely supported ones. Indeed, when the convex order dominating original probability measure is compactly supported, it is smaller than any of its dual quantizations while the dominated original measure is greater than any of its stationary (and therefore any of its quadratic optimal) primal quantization. Moreover, the quantization errors then correspond to martingale couplings between each original probability measure and its quantization. This permits to prove that any martingale coupling between the original probability measures can be approximated by a martingale coupling between their quantizations in Wassertein distance with a rate given by the quantization errors but also in the much finer adapted Wassertein distance. As a consequence, while the stability of (Weak) Martingale Optimal Transport problems with respect to the marginal distributions has only been established in dimension 1 so far, their value function computed numerically for the quantized marginals converges in any dimension to the value for the original probability measures as the numbers of quantization points go to ∞.
当用两个有限支持的概率测度逼近两个有序概率测度时,量化提供了一种非常自然的方法来保持凸阶。事实上,当紧支持凸阶支配原始概率测度时,它小于其任何对偶量化,而支配原始测度大于其任何平稳(因此大于其任何二次最优)原始量化。此外,量化误差对应于每个原始概率测度与其量化之间的鞅耦合。这允许证明原始概率测度之间的任何鞅耦合都可以通过它们在Wassertein距离中的量化之间的鞅耦合来近似,其速率由量化误差给定,但也可以在更精细地适应的Wassertein-distance中。因此,尽管到目前为止,(弱)鞅最优运输问题关于边缘分布的稳定性仅在维度1中建立,但随着量化点数达到∞,它们为量化边缘数值计算的值函数在任何维度上都收敛于原始概率测度的值。
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引用次数: 5
Fluctuations of the Magnetization for Ising models on Erdős-Rényi random graphs – the regimes of low temperature and external magnetic field Erdős-Rényi随机图上Ising模型磁化的波动——低温和外磁场的状态
IF 0.7 4区 数学 Q4 STATISTICS & PROBABILITY Pub Date : 2020-12-15 DOI: 10.30757/alea.v19-21
Z. Kabluchko, Matthias Lowe, K. Schubert
We continue our analysis of Ising models on the (directed) Erdős-Renyi random graph $G(N,p)$. We prove a quenched Central Limit Theorem for the magnetization and describe the fluctuations of the log-partition function. In the current note we consider the low temperature regime $beta>1$ and the case when an external magnetic field is present. In both cases, we assume that $p=p(N)$ satisfies $p^3N to infty$.
我们继续分析(有向)Erdős-Renyi随机图$G(N,p)$上的Ising模型。我们证明了磁化的一个淬灭中心极限定理,并描述了对数配分函数的涨落。在当前的说明中,我们考虑低温状态$beta>1$和存在外部磁场的情况。在这两种情况下,我们都假设$p=p(N)$满足$p^3N to infty$。
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引用次数: 4
Undirected Polymers in Random Environment: path properties in the mean field limit. 随机环境中的无向聚合物:平均场极限中的路径性质。
IF 0.7 4区 数学 Q4 STATISTICS & PROBABILITY Pub Date : 2020-12-07 DOI: 10.30757/alea.v19-61
N. Kistler, A. Schertzer
We consider the problem of undirected polymers (tied at the endpoints) in random environment, also known as the unoriented first passage percolation on the hypercube, in the limit of large dimensions. By means of the multiscale refinement of the second moment method we obtain a fairly precise geometrical description of optimal paths, i.e. of polymers with minimal energy. The picture which emerges can be loosely summarized as follows. The energy of the polymer is, to first approximation, uniformly spread along the strand. The polymer's bonds carry however a lower energy than in the directed setting, and are reached through the following geometrical evolution. Close to the origin, the polymer proceeds in oriented fashion -- it is thus as stretched as possible. The tension of the strand decreases however gradually, with the polymer allowing for more and more backsteps as it enters the core of the hypercube. Backsteps, although increasing the length of the strand, allow the polymer to connect reservoirs of energetically favorable edges which are otherwise unattainable in a fully directed regime. These reservoirs lie at mesoscopic distance apart, but in virtue of the high dimensional nature of the ambient space, the polymer manages to connect them through approximate geodesics with respect to the Hamming metric: this is the key strategy which leads to an optimal energy/entropy balance. Around halfway, the mirror picture sets in: the polymer tension gradually builds up again, until full orientedness close to the endpoint. The approach yields, as a corollary, a constructive proof of the result by Martinsson [Ann. Appl. Prob. 26 (2016), Ann. Prob. 46 (2018)] concerning the leading order of the ground state.
我们考虑随机环境中的无向聚合物(连接在端点)问题,也称为超立方体上的无向第一通道渗流,在大维极限下。通过二阶矩方法的多尺度精化,我们获得了最优路径(即具有最小能量的聚合物)的相当精确的几何描述。出现的情况可以大致概括如下。聚合物的能量,首先近似地,沿着链均匀地分布。然而,聚合物的键携带的能量低于定向设置中的能量,并通过以下几何演变达到。在接近原点的地方,聚合物以定向的方式进行,从而尽可能地拉伸。然而,链的张力逐渐降低,当聚合物进入超立方体的核心时,它允许越来越多的反跳。后台阶虽然增加了链的长度,但允许聚合物连接具有能量有利边缘的储层,否则在完全定向的状态下是无法实现的。这些储层相距介观距离,但由于环境空间的高维性质,聚合物设法通过关于汉明度量的近似测地线将它们连接起来:这是导致最佳能量/熵平衡的关键策略。大约在中途,镜像开始显现:聚合物张力逐渐再次建立,直到接近终点时完全定向。作为推论,该方法得出了Martinsson[Ann.Appl.Prob.26(2016),Ann.Prob.46(2018)]关于基态主导序的结果的构造性证明。
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引用次数: 1
Strong laws of large numbers for a growth-fragmentation process with bounded cell sizes 细胞大小有界的生长碎裂过程的强数定律
IF 0.7 4区 数学 Q4 STATISTICS & PROBABILITY Pub Date : 2020-12-06 DOI: 10.30757/ALEA.v19-68
E. Horton, A. Watson
Growth-fragmentation processes model systems of cells that grow continuously over time and then fragment into smaller pieces. Typically, on average, the number of cells in the system exhibits asynchronous exponential growth and, upon compensating for this, the distribution of cell sizes converges to an asymptotic profile. However, the long-term stochastic behaviour of the system is more delicate, and its almost sure asymptotics have been so far largely unexplored. In this article, we study a growth-fragmentation process whose cell sizes are bounded above, and prove the existence of regimes with differing almost sure long-term behaviour.
生长-分裂过程模拟细胞系统,随着时间的推移不断生长,然后分裂成更小的碎片。通常,平均而言,系统中的细胞数量呈现异步指数增长,并且在补偿这一点后,细胞大小的分布收敛于渐近轮廓。然而,该系统的长期随机行为更为微妙,其几乎肯定的渐近性迄今在很大程度上尚未得到探索。在本文中,我们研究了一个细胞大小有界的生长-分裂过程,并证明了具有不同的几乎确定的长期行为的制度的存在。
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引用次数: 1
Two-sided immigration, emigration and symmetry properties of self-similar interval partition evolutions 自相似区间划分演化的双侧迁移、迁移和对称性
IF 0.7 4区 数学 Q4 STATISTICS & PROBABILITY Pub Date : 2020-11-26 DOI: 10.30757/alea.v20-25
Quan Shi, Matthias Winkel
Forman et al. (2020+) constructed $(alpha,theta)$-interval partition evolutions for $alphain(0,1)$ and $thetage 0$, in which the total sums of interval lengths ("total mass") evolve as squared Bessel processes of dimension $2theta$, where $thetage 0$ acts as an immigration parameter. These evolutions have pseudo-stationary distributions related to regenerative Poisson--Dirichlet interval partitions. In this paper we study symmetry properties of $(alpha,theta)$-interval partition evolutions. Furthermore, we introduce a three-parameter family ${rm SSIP}^{(alpha)}(theta_1,theta_2)$ of self-similar interval partition evolutions that have separate left and right immigration parameters $theta_1ge 0$ and $theta_2ge 0$. They also have squared Bessel total mass processes of dimension $2theta$, where $theta=theta_1+theta_2-alphage-alpha$ covers emigration as well as immigration. Under the constraint $max{theta_1,theta_2}gealpha$, we prove that an ${rm SSIP}^{(alpha)}(theta_1,theta_2)$-evolution is pseudo-stationary for a new distribution on interval partitions, whose ranked sequence of lengths has Poisson--Dirichlet distribution with parameters $alpha$ and $theta$, but we are unable to cover all parameters without developing a limit theory for composition-valued Markov chains, which we do in a sequel paper.
Forman等人(2020+)构建了$alphain(0,1)$和$thetage 0$的$(alpha,theta)$ -区间划分演化,其中区间长度的总和(“总质量”)演化为维度$2theta$的平方贝塞尔过程,其中$thetage 0$作为迁移参数。这些演化具有与再生泊松—狄利克雷区间划分相关的伪平稳分布。本文研究了$(alpha,theta)$ -区间划分演化的对称性。此外,我们还引入了一个三参数族${rm SSIP}^{(alpha)}(theta_1,theta_2)$的自相似区间划分演化,该演化具有独立的左右迁移参数$theta_1ge 0$和$theta_2ge 0$。他们也有平方贝塞尔总质量过程的维度$2theta$,其中$theta=theta_1+theta_2-alphage-alpha$包括迁出和迁入。在约束$max{theta_1,theta_2}gealpha$下,我们证明了区间分区上的一个新分布的${rm SSIP}^{(alpha)}(theta_1,theta_2)$ -演化是伪平稳的,该分布的排序长度序列具有泊松—狄利克雷分布,参数为$alpha$和$theta$,但是我们不能在不发展复合值马尔可夫链的极限理论的情况下覆盖所有的参数,我们在后续论文中做了。
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引用次数: 3
Extinction probabilities in branching processes with countably many types: a general framework 具有可数多类型分支过程的灭绝概率:一般框架
IF 0.7 4区 数学 Q4 STATISTICS & PROBABILITY Pub Date : 2020-11-19 DOI: 10.30757/alea.v19-12
D. Bertacchi, Peter Braunsteins, S. Hautphenne, F. Zucca
We consider Galton-Watson branching processes with countable typeset $mathcal{X}$. We study the vectors ${bf q}(A)=(q_x(A))_{xinmathcal{X}}$ recording the conditional probabilities of extinction in subsets of types $Asubseteq mathcal{X}$, given that the type of the initial individual is $x$. We first investigate the location of the vectors ${bf q}(A)$ in the set of fixed points of the progeny generating vector and prove that $q_x({x})$ is larger than or equal to the $x$th entry of any fixed point, whenever it is different from 1. Next, we present equivalent conditions for $q_x(A)< q_x (B)$ for any initial type $x$ and $A,Bsubseteq mathcal{X}$. Finally, we develop a general framework to characterise all emph{distinct} extinction probability vectors, and thereby to determine whether there are finitely many, countably many, or uncountably many distinct vectors. We illustrate our results with examples, and conclude with open questions.
我们考虑具有可数类型集$mathcal{X}$的Galton Watson分支过程。我们研究了向量${bf-q}(A)=(q_x(A))_{xinmathcal{x}}$,记录了类型$Asubsteqmathcal{x}$的子集中灭绝的条件概率,假定初始个体的类型为$x$。我们首先研究向量${bf-q}(A)$在子代生成向量的不动点集中的位置,并证明$q_x({x})$大于或等于任何不动点的第$x$个入口,只要它不同于1。接下来,我们给出了任何初始类型$x$和$A,Bsubsteqmathcal{x}$的$q_x(A)
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引用次数: 2
Fractionally Integrated Moving Average Stable Processes With Long-Range Dependence 具有长程依赖性的分数积分移动平均稳定过程
IF 0.7 4区 数学 Q4 STATISTICS & PROBABILITY Pub Date : 2020-11-11 DOI: 10.30757/alea.v19-23
G. Feltes, S. Lopes
Long memory processes driven by Levy noise with finite second-order moments have been well studied in the literature. They form a very rich class of processes presenting an autocovariance function which decays like a power function. Here, we study a class of Levy process whose second-order moments are infinite, the so-called $alpha$-stable processes. Based on Samorodnitsky and Taqqu (2000), we construct an isometry that allows us to define stochastic integrals concerning the linear fractional stable motion using Riemann-Liouville fractional integrals. With this construction, follows naturally an integration by parts formula. We then present a family of stationary $Salpha S$ processes with the property of long-range dependence, using a generalized measure to investigate its dependence structure. In the end, the law of large number's result for a time's sample of the process is shown as an application of the isometry and integration by parts formula.
由具有有限二阶矩的Levy噪声驱动的长记忆过程在文献中已经得到了很好的研究。它们形成了一类非常丰富的过程,呈现出像幂函数一样衰减的自协方差函数。在这里,我们研究了一类二阶矩无穷大的Levy过程,即所谓的$alpha$稳定过程。基于Samorodnitsky和Taqqu(2000),我们构造了一个等距图,使我们能够使用Riemann-Liouville分数积分定义与线性分数稳定运动有关的随机积分。有了这种结构,自然就遵循了部分积分公式。然后,我们提出了一类具有长程依赖性质的平稳$Salpha-S$过程,并使用一个广义测度来研究其依赖结构。最后,将过程中一个时间样本的大数结果定律作为等距和分部积分公式的一个应用加以说明。
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引用次数: 0
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Alea-Latin American Journal of Probability and Mathematical Statistics
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