Theory of Probability and Mathematical Statistics最新文献

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On local path behavior of Surgailis multifractional processes 多分数过程的局部路径行为

IF 0.9 Q4 STATISTICS & PROBABILITY

Theory of Probability and Mathematical Statistics

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 <italic>classical</italic> Multifractional Brownian Motion (MBM) <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace script upper M left-parenthesis t right-parenthesis right-brace Subscript t element-of double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{{mathcal {M}}(t)}_{tin mathbb {R}}</mml:annotation> </mml:semantics></mml:math></inline-formula> of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{mathcal {H}}</mml:annotation> </mml:semantics></mml:math></inline-formula> of the well-known Fractional Brownian Motion by a deterministic function <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{mathcal {H}}(t)</mml:annotation> </mml:semantics></mml:math></inline-formula> 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 <italic>non-classical</italic> 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 Q4 STATISTICS & PROBABILITY

Theory of Probability and Mathematical Statistics

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 <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Z(x)</mml:annotation> </mml:semantics></mml:math></inline-formula> and construct approximations of solutions <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">U(x)</mml:annotation> </mml:semantics></mml:math></inline-formula> of ordinary or partial differential equations whose random coefficients depend on <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Z(x)</mml:annotation> </mml:semantics></mml:math></inline-formula>. FD models of <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Z(x)</mml:annotation> </mml:semantics></mml:math></inline-formula> and <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">U(x)</mml:annotation> </mml:semantics></mml:math></inline-formula> 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 <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z left-parenthes

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 Q4 STATISTICS & PROBABILITY

Theory of Probability and Mathematical Statistics

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。

引用次数: 0

Multivariate Gaussian Random Fields over Generalized Product Spaces involving the Hypertorus 涉及超环面的广义积空间上的多元高斯随机场

IF 0.9 Q4 STATISTICS & PROBABILITY

Theory of Probability and Mathematical Statistics

Pub Date : 2022-02-22 DOI: 10.1090/tpms/1176

F. Bachoc, A. Peron, E. Porcu

The paper deals with multivariate Gaussian random fields defined over generalized product spaces that involve the hypertorus. The assumption of Gaussianity implies the finite dimensional distributions to be completely specified by the covariance functions, being in this case matrix valued mappings.We start by considering the spectral representations that in turn allow for a characterization of such covariance functions. We then provide some methods for the construction of these matrix valued mappings. Finally, we consider strategies to evade radial symmetry (called isotropy in spatial statistics) and provide representation theorems for such a more general case.

本文讨论了在广义乘积空间上定义的多变量高斯随机场，它涉及超轨道。高斯性的假设意味着有限维分布完全由协方差函数指定，在这种情况下是矩阵值映射。我们首先考虑频谱表示，这反过来又允许对这种协方差函数进行表征。然后，我们提供了一些构造这些矩阵值映射的方法。最后，我们考虑了避免径向对称（在空间统计学中称为各向同性）的策略，并为这种更普遍的情况提供了表示定理。

引用次数: 1

Boltzmann–Gibbs Random Fields with Mesh-free Precision Operators Based on Smoothed Particle Hydrodynamics 基于光滑粒子流体力学的无网格精度算子Boltzmann-Gibbs随机场

IF 0.9 Q4 STATISTICS & PROBABILITY

Theory of Probability and Mathematical Statistics

Pub Date : 2022-01-26 DOI: 10.1090/tpms/1180

D. Hristopulos

Boltzmann–Gibbs random fields are defined in terms of the exponential expression exp ⁡ ( − H ) exp left (-mathcal {H}right ) , where H mathcal {H} is a suitably defined energy functional of the field states x ( s ) x(mathbf {s}) . This paper presents a new Boltzmann–Gibbs model which features local interactions in the energy functional. The interactions are embodied in a spatial coupling function which uses smoothed kernel-function approximations of spatial derivatives inspired from the theory of smoothed particle hydrodynamics. A specific model for the interactions based on a second-degree polynomial of the Laplace operator is studied. An explicit, mesh-free expression of the spatial coupling function (precision function) is derived for the case of the squared exponential (Gaussian) smoothing kernel. This coupling function allows the model to seamlessly extend from discrete data vectors to continuum fields. Connections with Gaussian Markov random fields and the Matérn field with ν = 1 nu =1 are established.

玻尔兹曼-吉布斯随机场用指数表达式exp (- H) expleft (- mathcal H{}right)来定义，其中H mathcal H{是场态x(s) x(}mathbf s{)的适当定义的能量函数。本文提出了一种新的具有能量泛函局部相互作用的玻尔兹曼-吉布斯模型。相互作用体现在一个空间耦合函数中，该函数使用了受光滑粒子流体力学理论启发的空间导数的光滑核函数近似。研究了一种基于拉普拉斯算子二阶多项式的特定模型。对于平方指数(高斯)平滑核，导出了空间耦合函数(精度函数)的显式无网格表达式。这种耦合功能允许模型从离散数据向量无缝扩展到连续域。建立了高斯马尔可夫随机场与ν =1 }nu =1的mat rn场的连接。

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引用次数: 0

Series representations and simulations of isotropic random fields in the Euclidean space 欧氏空间中各向同性随机场的级数表示与模拟

IF 0.9 Q4 STATISTICS & PROBABILITY

Theory of Probability and Mathematical Statistics

Pub Date : 2021-12-07 DOI: 10.1090/tpms/1158

Z. Ma, C. Ma

This paper introduces the series expansion for homogeneous, isotropic and mean square continuous random fields in the Euclidean space, which involves the Bessel function and the ultraspherical polynomial, but differs from the spectral representation in terms of the ordinary spherical harmonics that has more terms at each level.The series representation provides a simple and efficient approach for simulation of isotropic (non-Gaussian) random fields.

本文介绍了欧氏空间中齐次、各向同性和均方连续随机场的级数展开，它涉及贝塞尔函数和超球面多项式，但与谱表示不同的是，在每一级都有更多项的普通球面谐波。级数表示为各向同性（非高斯）随机场的模拟提供了一种简单有效的方法。

引用次数: 0

On the least squares estimator asymptotic normality of the multivariate symmetric textured surface parameters 多元对称纹理曲面参数的最小二乘估计渐近正态性

IF 0.9 Q4 STATISTICS & PROBABILITY

Theory of Probability and Mathematical Statistics

Pub Date : 2021-12-07 DOI: 10.1090/tpms/1161

A. Ivanov, I. Savych

A multivariate trigonometric regression model is considered. Various discrete modifications of the similar bivariate model received serious attention in the literature on signal and image processing due to multiple applications in the analysis of symmetric textured surfaces. In the paper asymptotic normality of the least squares estimator for amplitudes and angular frequencies is obtained in multivariate trigonometric model assuming that the random noise is a homogeneous or homogeneous and isotropic Gaussian, in particular, strongly dependent random field on R M , M > 2. mathbb {R}^M,,, M>2.

考虑了一个多元三角回归模型。由于在对称纹理表面分析中的多种应用，类似的二元模型的各种离散修改在信号和图像处理的文献中受到了重视。本文在多元三角模型中，假设随机噪声是齐次或齐次各向同性高斯，特别是R M，M>2上的强相关随机场，得到了振幅和角频率的最小二乘估计的渐近正态性。mathbb｛R｝^M，，，M>2。

引用次数: 3

Convergence in distribution for randomly stopped random fields 随机停止随机场分布的收敛性

IF 0.9 Q4 STATISTICS & PROBABILITY

Theory of Probability and Mathematical Statistics

Pub Date : 2021-12-07 DOI: 10.1090/tpms/1160

D. Silvestrov

Let <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper X"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathbb {X}</mml:annotation> </mml:semantics></mml:math></inline-formula> and <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Y"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathbb {Y}</mml:annotation> </mml:semantics></mml:math></inline-formula> be two complete, separable, metric spaces, <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi Subscript epsilon Baseline left-parenthesis x right-parenthesis comma x element-of double-struck upper X"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">X</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">xi _varepsilon (x), x in mathbb {X}</mml:annotation> </mml:semantics></mml:math></inline-formula> and <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu Subscript epsilon"> <mml:semantics> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>ε</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">nu _varepsilon</mml:annotation> </mml:semantics></mml:math></inline-formula> be, for every <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon element-of left-bracket 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">varepsilon in [0, 1]</mml:annotation> </mml:semantics></mml:math></inline-formula>, respectively, a random field taking values in space <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Y"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/

设X mathbb X{和Y }mathbb Y{是两个完备的，可分离的度量空间，ξ ε (X)， X∈X }xi ＿ varepsilon (X)， X inmathbb X{和ν ε }nu ＿ varepsilon be，对于每一个ε∈[0]，1] varepsilonin[0,1]分别为在空间Y中取值的随机场mathbb Y{和在空间X中取值的随机变量}mathbb X{。给出了随机变量ξ ε (ν ε) }xi ＿ varepsilon (nu ＿ varepsilon)分布收敛的一般条件，即保证关系成立的条件。ξ ε (ν ε) δ ξ 0(ν 0) xi ＿ varepsilon (nu ＿ varepsilon) stackrel{mathsf d{}}{longrightarrow}xi ＿0(nu ＿0)为ε→0varepsilonto

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引用次数: 0

On the locations of maxima and minima in a sequence of exchangeable random variables 关于可交换随机变量序列中极大值和极小值的位置

IF 0.9 Q4 STATISTICS & PROBABILITY

Theory of Probability and Mathematical Statistics

Pub Date : 2021-12-07 DOI: 10.1090/tpms/1154

D. Ferger

We show for a finite sequence of exchangeable random variables that the locations of the maximum and minimum are independent from every symmetric event. In particular they are uniformly distributed on the grid without the diagonal. Moreover, for an infinite sequence we show that the extrema and their locations are asymptotically independent. Here, in contrast to the classical approach we do not use affine-linear transformations. Moreover it is shown how the new transformations can be used in extreme value statistics.

我们证明了对于可交换随机变量的有限序列，最大值和最小值的位置与每个对称事件无关。特别是，它们在没有对角线的网格上均匀分布。此外，对于无穷序列，我们证明了极值及其位置是渐近独立的。这里，与经典方法相反，我们不使用仿射线性变换。此外，还展示了如何在极值统计中使用新的转换。

引用次数: 0

For which functions are 𝑓(𝑋_{𝑡})-𝔼𝕗(𝕏_{𝕥}) and 𝕘(𝕏_{𝕥})/𝔼𝕘(𝕏_{𝕥}) martingales?

IF 0.9 Q4 STATISTICS & PROBABILITY

Theory of Probability and Mathematical Statistics

Pub Date : 2021-12-07 DOI: 10.1090/tpms/1157

F. Kühn, R. Schilling

Let <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X equals left-parenthesis upper X Subscript t Baseline right-parenthesis Subscript t greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">X=(X_t)_{tgeq 0}</mml:annotation> </mml:semantics></mml:math></inline-formula> be a one-dimensional Lévy process such that each <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Subscript t"> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">X_t</mml:annotation> </mml:semantics></mml:math></inline-formula> has a <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript b Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">C^1_b</mml:annotation> </mml:semantics></mml:math></inline-formula>-density w. r. t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon double-struck upper R right-arrow double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">fcolon mathbb {R}to mathbb {R}</mml:annotation> </mml:semantics></mml:math></inline-formula>, and exponentially bounded functions <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g colon double-struck upper R right-arrow left-parenthesis 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <m

设X=(X t) t≥0 X=(X＿t){＿tgeq 0为}一维lsamvy过程，使得每个X t X＿t具有cb1 C^1＿b -密度w. r. t.勒贝格测度和某些多项式或指数矩。我们描述了所有多项式有界函数f: R→R fcolonmathbb R{}tomathbb R{，以及指数有界函数g:R→(0，∞)g }colonmathbb R{}to (0, infty)，使得f(X t)−E f(X t) f(X＿t)- mathbb E{ f(X＿t)g(X t)/ eg (X t) g(X＿t)/ }mathbb eg (X＿t)是鞅。{}

{"title":"For which functions are 𝑓(𝑋_{𝑡})-𝔼𝕗(𝕏_{𝕥}) and 𝕘(𝕏_{𝕥})/𝔼𝕘(𝕏_{𝕥}) martingales?","authors":"F. Kühn, R. Schilling","doi":"10.1090/tpms/1157","DOIUrl":"https://doi.org/10.1090/tpms/1157","url":null,"abstract":"Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X equals left-parenthesis upper X Subscript t Baseline right-parenthesis Subscript t greater-than-or-equal-to 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mi>t</mml:mi>\u0000 </mml:msub>\u0000 <mml:msub>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo>≥</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">X=(X_t)_{tgeq 0}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a one-dimensional Lévy process such that each <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X Subscript t\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mi>t</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">X_t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> has a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Subscript b Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msubsup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mi>b</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msubsup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^1_b</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-density w. r. t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon double-struck upper R right-arrow double-struck upper R\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo>:</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">→</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">fcolon mathbb {R}to mathbb {R}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, and exponentially bounded functions <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g colon double-struck upper R right-arrow left-parenthesis 0 comma normal infinity right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>g</mml:mi>\u0000 <mml:mo>:</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">→</mml:mo>\u0000 <m","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43522090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

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