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IF 0.9 Q3 Mathematics Pub Date : 2022-11-08 DOI: 10.1090/tpms/1183
A. Malyarenko, Y. Mishura, A. Olenko, M. Ostoja-Starzewski
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引用次数: 0
Revisiting recurrence criteria of birth and death processes. Short proofs 重新审视出生和死亡过程的复发标准。简短的证明
IF 0.9 Q3 Mathematics Pub Date : 2022-11-08 DOI: 10.1090/tpms/1182
O. Zakusylo
The paper contains several new transparent proofs of criteria appearing in classification of birth and death processes (BDPs). They are almost purely probabilistic and differ from the classical techniques of three-term recurrence relations, continued fractions and orthogonal polynomials. Let T ∞ {T^infty } be the passage time from zero to ∞ infty . The regularity criterion says that T ∞ > ∞ {T^infty } > infty if and only if E T ∞ > ∞ mathbb {E}{T^infty } > infty . It is heavily based on a result of Gong, Y., Mao, Y.-H. and Zhang, C. [J. Theoret. Probab. 25 (2012), no. 4, 950–980]. We obtain the latter expectation by using a two-term recurrence relation. We observe that the recurrence criterion is an immediate consequence of the well-known recurrence criterion for discrete-time BDPs and a result of Chung K. L. [Markov Chains with Stationary Transition Probabilities, Springer-Verlag, New York (1967)]. We obtain the classical criterion of positive recurrence using technique of the common probability space. While doing so, we construct a monotone sequence of BDPs with finite state spaces converging to BDPs with an infinite state space.
本文包含了出生和死亡过程分类标准的几个新的透明证明。它们几乎是纯概率的,不同于三项递推关系、连分式和正交多项式的经典技术。设T∞{T^infty}为从零到∞infty的通过时间。正则性准则表明,T∞>∞{T^infty}>infty当且仅当E T∞>∞mathbb{E}{T^ infty}>infty。这在很大程度上是基于龚、毛和张的一个结果。[J.Theoret.Probab.252012,no.4950-980]。我们通过使用两项递推关系得到了后一种期望。我们观察到,递推准则是众所周知的离散时间BDP递推准则的直接结果,也是Chung K.L.[具有平稳转移概率的马尔可夫链,Springer Verlag,纽约(1967)]的结果。利用公共概率空间技术,得到了正递推的经典判据。在这样做的同时,我们构造了一个具有有限状态空间的BDP的单调序列,该序列收敛于具有无限状态空间的BD。
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引用次数: 0
Isotropic random spin weighted functions on 𝑆² vs isotropic random fields on 𝑆³
IF 0.9 Q3 Mathematics Pub Date : 2022-11-08 DOI: 10.1090/tpms/1177
Michele Stecconi

We show that an isotropic random field on S U ( 2 ) SU(2) is not necessarily isotropic as a random field on S 3 S^3 , although the two spaces can be identified. The ambiguity is due to the fact that the notion of isotropy on a group and on a sphere are different, the latter being much stronger. We show that any isotropic random field on S 3 S^3 is necessarily a superposition of uncorrelated random harmonic homogeneous polynomials, such that the one of degree d d is necessarily a superposition of uncorrelated random spin weighted functions of every possible spin weight in the range { d 2 , , d 2 } bigl {

我们证明了SU(2)SU(2。这种模糊性是由于群和球体上的各向同性概念不同,后者更强。我们证明了在S3 S^3上的任何各向同性随机场必然是不相关的随机调和齐次多项式的叠加,使得次数为d的一个必然是在{−d2,…,d2}bigl{-frac{d}{2},dots范围内的每个可能的自旋权重的不相关随机自旋加权函数的叠加,frac{d}{2}bigr},它们中的每一个在SU(2)SU(2)的意义上是各向同性的。此外,对于固定度的随机场,在某种意义上,每个自旋权重都以相同的大小出现。此外,我们还将概述自旋加权函数和Wigner D-矩阵的理论,目的是收集许多不同的观点并添加我们的观点。作为这项研究的副产品,我们将证明Wigner矩阵的一些新性质,以及一个关于算子的公式→ S 2 S^3到S^2,在[Bérard Bergery和Bourguignon,Illinois J.Math.26(1982),no.2181-200]的意义上
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引用次数: 0
Spatiotemporal covariance functions for Laplacian ARMA fields in higher dimensions 高维拉普拉斯ARMA场的时空协方差函数
IF 0.9 Q3 Mathematics Pub Date : 2022-11-08 DOI: 10.1090/tpms/1173
G. Terdik
This paper presents clear formulae of the covariance functions of Laplacian ARMA fields in terms of coefficients and Bessel functions in higher spatial dimensions. Spectral methods are used for the study of spatiotemporal Laplacian ARMA fields in Euclidean spaces and spheres therein with dimension d ≥ 2 dgeq 2 .
本文给出了拉普拉斯ARMA场的协方差函数在高空间维度上的系数公式和贝塞尔函数公式。用谱方法研究了欧氏空间中的时空拉普拉斯ARMA场及其维数d≥2dgeq2的球面。
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引用次数: 1
On spectral theory of random fields in the ball 球中随机场的谱理论
IF 0.9 Q3 Mathematics Pub Date : 2022-11-08 DOI: 10.1090/tpms/1175
N. Leonenko, A. Malyarenko, A. Olenko
The paper investigates random fields in the ball. It studies three types of such fields: restrictions of scalar random fields in the ball to the sphere, spin, and vector random fields. The review of the existing results and new spectral theory for each of these classes of random fields are given. Examples of applications to classical and new models of these three types are presented. In particular, the Matérn model is used for illustrative examples. The derived spectral representations can be utilised to further study theoretical properties of such fields and to simulate their realisations. The obtained results can also find various applications for modelling and investigating ball data in cosmology, geosciences and embryology.
本文研究了球中的随机场。它研究了三种类型的此类场:球中标量随机场对球体的限制、自旋和矢量随机场。对每一类随机场的现有结果和新的谱理论进行了综述。给出了这三种类型的经典模型和新模型的应用实例。特别是,Matérn模型用于举例说明。导出的光谱表示可用于进一步研究此类场的理论性质并模拟其实现。所获得的结果还可以在宇宙学、地球科学和胚胎学中用于建模和研究球体数据。
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引用次数: 3
On the other LIL for variables without finite variance 在另一个LIL上,对于没有有限方差的变量
IF 0.9 Q3 Mathematics Pub Date : 2022-11-08 DOI: 10.1090/tpms/1179
R. Pakshirajan, M. Sreehari

In this paper we give a simpler proof of Jain’s [Z. Wahrsch. Verw. Gebiete 59 (1982), no. 1, 117–138] result concerning the Other Law of the Iterated Logarithm for partial sums of a class of independent and identically distributed random variables with infinite variance but in the domain of attraction of a normal law. Jain’s result is less restrictive than ours but depends heavily on the techniques of Donsker and Varadhan in the theory of Large deviations. Our proof involves elementary properties of slowly varying functions. We assume that the distribution of random variables X n X_n satisfies the condition that lim x log H ( x ) ( log x ) δ = 0 lim _{ xrightarrow infty } frac {log H(x)}{(log x)^delta } = 0 for some 0 > δ > 1 / 2

在本文中,我们给出了Jain的[Z.Warsch.Verw.Gebiete 59(1982),no.1117–138]关于一类方差无穷但在正态律吸引域中的独立同分布随机变量的部分和的重对数另一定律的结果的一个更简单的证明。Jain的结果没有我们的结果那么严格,但在很大程度上取决于Donsker和Varadhan在大偏差理论中的技术。我们的证明涉及慢变函数的基本性质。我们假定随机变量XnX_n的分布满足lim→ ∞ 日志⁡ H(x)(对数⁡ x)δ=0 lim _{xrightarrowinfty}frac{log H(x)}{(log x)^delta}=0,其中H(x)=E(x 12 I(|x 1|≤x))H(x)=mathsf Eleft(x_1^2 I(|x_1|le x)right)是一个缓慢变化的函数。上述条件限制性不强。
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引用次数: 0
Stationary solutions of a second-order differential equation with operator coefficients 一类具有算子系数的二阶微分方程的平稳解
IF 0.9 Q3 Mathematics Pub Date : 2022-05-16 DOI: 10.1090/tpms/1171
M. Horodnii
Necessary and sufficient conditions are given for the existence of a unique stationary solution to the second-order linear differential equation with bounded operator coefficients, perturbed by a stationary process. In the case when the corresponding “algebraic” operator equation has separated roots, the new representation of the stationary solution of the considered differential equation is obtained.
给出了算子系数有界的二阶线性微分方程在平稳过程扰动下存在唯一平稳解的充要条件。在相应的“代数”算子方程具有分离根的情况下,得到了所考虑的微分方程平稳解的新表示。
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引用次数: 0
On local path behavior of Surgailis multifractional processes 多分数过程的局部路径行为
IF 0.9 Q3 Mathematics Pub Date : 2022-05-16 DOI: 10.1090/tpms/1162
A. Ayache, F. Bouly

Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the classical Multifractional Brownian Motion (MBM) { M ( t ) } t R {{mathcal {M}}(t)}_{tin mathbb {R}} of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter H {mathcal {H}} of the well-known Fractional Brownian Motion by a deterministic function H ( t ) {mathcal {H}}(t) having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two non-classical Gaussian multi

多分数过程是具有非平稳增量的随机过程,其局部正则性和自相似性随点而变化。典型的例子是经典的多分数布朗运动(MBM) {M (t)} t∈R {{mathcal {M}}(t)}_{tin mathbb {R}} (Benassi, Jaffard, lsamvy vsamhel, Peltier和Roux),它是在90年代中期建立的,只是用一个具有一定平滑性的确定性函数H (t) {mathcal {H}}(t)代替了著名的分数阶布朗运动中的常数Hurst参数H {mathcal {H}}。十多年后,使用一种不同的构造方法,基本上依赖于非齐次分数阶积分和微分算子,Surgailis引入了两个非经典高斯多阶过程,表示为{X(t)} t∈R {X(t)}_{tin mathbb {R}}和{Y(t)} t∈R {Y(t)}_{tin mathbb {R}}。在函数参数H(⋅){mathcal {H}}(cdot)的较弱条件下,我们证明了{M (t)} t∈R {{mathcal {M}}(t)}_{tin mathbb {R}}和{X(t)} t∈R {X(t)}_{tin mathbb {R}}以及{M (t)}} t∈R {{mathcal {M}}(t)}_{tin mathbb {R}}和{Y(t)} t∈R {Y(t)}_{tin mathbb {R}}只有局部的区别
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引用次数: 1
Finite dimensional models for random microstructures 随机微观结构的有限维模型
IF 0.9 Q3 Mathematics Pub Date : 2022-05-16 DOI: 10.1090/tpms/1168
M. Grigoriu

Finite dimensional (FD) models, i.e., deterministic functions of space depending on finite sets of random variables, are used extensively in applications to generate samples of random fields Z ( x ) Z(x) and construct approximations of solutions U ( x ) U(x) of ordinary or partial differential equations whose random coefficients depend on Z ( x ) Z(x) . FD models of Z ( x ) Z(x) and U ( x ) U(x) constitute surrogates of these random fields which target various properties, e.g., mean/correlation functions or sample properties. We establish conditions under which samples of FD models can be used as substitutes for samples of

有限维(FD)模型,即取决于随机变量的有限集合的空间的确定性函数,在应用中被广泛地用于生成随机场Z(x)Z(x)的样本,并构造其随机系数取决于Z(x。Z(x)Z(x)和U(x)U(x)的FD模型构成了这些随机场的替代物,这些随机场以各种性质为目标,例如,均值/相关函数或样本性质。我们建立了FD模型的样本可以用作两种类型的随机场Z(x)Z(x)和一个简单随机方程的Z(x,Z(x,Z)和U(x)U(x)样本的替代品的条件。其中一些条件通过数值例子加以说明。
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引用次数: 3
Non-central limit theorem for the spatial average of the solution to the wave equation with Rosenblatt noise 含Rosenblatt噪声的波动方程解的空间平均的非中心极限定理
IF 0.9 Q3 Mathematics Pub Date : 2022-05-16 DOI: 10.1090/tpms/1167
R. Dhoyer, C. Tudor
We analyze the limit behavior in distribution of the spatial average of the solution to the wave equation driven by the two-parameter Rosenblatt process in spatial dimension d = 1 d=1 . We prove that this spatial average satisfies a non-central limit theorem, more precisely it converges in law to a Wiener integral with respect to the Rosenblatt process. We also give a functional version of this limit theorem.
本文分析了双参数Rosenblatt过程驱动的波动方程在空间维数d=1时的空间平均解的极限分布行为。我们证明了该空间平均满足一个非中心极限定理,更确切地说,它在定律上收敛于关于Rosenblatt过程的Wiener积分。我们也给出了这个极限定理的泛函形式。
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引用次数: 0
期刊
Theory of Probability and Mathematical Statistics
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