We consider the distribution of the product of a Wishart matrix and a normal vector with uncommon covariance matrices. We derive the stochastic representation which reduces the computational burden for the generation of realizations of the product. Using this representation, the density function and higher order moments of the product are derived. In a numerical illustration, we investigate some properties of the distribution of the product. We further suggest the Edgeworth type expansions for the product, and we observe that the suggested approximations provide a good performance for moderately large degrees of freedom of a Wishart matrix.
{"title":"Distribution of the product of a Wishart matrix and a normal vector","authors":"Koshiro Yonenaga, A. Suzukawa","doi":"10.1090/tpms/1193","DOIUrl":"https://doi.org/10.1090/tpms/1193","url":null,"abstract":"We consider the distribution of the product of a Wishart matrix and a normal vector with uncommon covariance matrices. We derive the stochastic representation which reduces the computational burden for the generation of realizations of the product. Using this representation, the density function and higher order moments of the product are derived. In a numerical illustration, we investigate some properties of the distribution of the product. We further suggest the Edgeworth type expansions for the product, and we observe that the suggested approximations provide a good performance for moderately large degrees of freedom of a Wishart matrix.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47755032","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}
<p>In this note we prove some sufficient conditions for transience and recurrence of a Lévy-type process in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, whose generator defined on the test functions is of the form <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L f left-parenthesis x right-parenthesis equals integral Underscript double-struck upper R Endscripts left-parenthesis f left-parenthesis x plus u right-parenthesis minus f left-parenthesis x right-parenthesis minus nabla f left-parenthesis x right-parenthesis dot u double-struck 1 Subscript StartAbsoluteValue u EndAbsoluteValue less-than-or-equal-to 1 Baseline right-parenthesis nu left-parenthesis x comma d u right-parenthesis comma f element-of upper C Subscript normal infinity Superscript 2 Baseline left-parenthesis double-struck upper R right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant="normal">∇<!-- ∇ --></mml:mi> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn mathvariant="double-struck">1</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow>
在本文中，我们证明了R mathbb R{中l型过程的暂态和递归的几个充分条件。其生成函数定义在测试函数上的形式为L f (x) =∫R (f (x + u) - f (x) -∇f (x)·u 1 | u |≤1)ν (x, du)， f∈C∞2 (R)。}begin{equation*} Lf(x) =int _{mathbb {R}} left ( f(x+u)-f(x)- nabla f(x)cdot u mathbb {1}_{|u|leq 1} right ) nu (x,du), quad fin C_infty ^2(mathbb {R}). end{equation*}这里的ν (x,du) nu (x,du)是一个l型核，它的尾部要么有规律地扩展变化，要么衰减得足够快。为了证明，使用了Foster-Lyapunov方法。
{"title":"On recurrence and transience of some Lévy-type processes in ℝ","authors":"V. Knopova","doi":"10.1090/tpms/1187","DOIUrl":"https://doi.org/10.1090/tpms/1187","url":null,"abstract":"<p>In this note we prove some sufficient conditions for transience and recurrence of a Lévy-type process in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {R}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, whose generator defined on the test functions is of the form <disp-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L f left-parenthesis x right-parenthesis equals integral Underscript double-struck upper R Endscripts left-parenthesis f left-parenthesis x plus u right-parenthesis minus f left-parenthesis x right-parenthesis minus nabla f left-parenthesis x right-parenthesis dot u double-struck 1 Subscript StartAbsoluteValue u EndAbsoluteValue less-than-or-equal-to 1 Baseline right-parenthesis nu left-parenthesis x comma d u right-parenthesis comma f element-of upper C Subscript normal infinity Superscript 2 Baseline left-parenthesis double-struck upper R right-parenthesis period\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:msub>\u0000 <mml:mo>∫<!-- ∫ --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mrow>\u0000 <mml:mo>(</mml:mo>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>⋅<!-- ⋅ --></mml:mo>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn mathvariant=\"double-struck\">1</mml:mn>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mo>≤<!-- ≤ --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 ","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45133077","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}
The paper is devoted to three-parametric self-similar Gaussian Volterra processes that generalize fractional Brownian motion. We study the asymptotic growth of such processes and the properties of long- and short-range dependence. Then we consider the problem of the drift parameter estimation for Ornstein–Uhlenbeck process driven by Gaussian Volterra process under consideration. We construct a strongly consistent estimator and investigate its asymptotic properties. Namely, we prove that it has the Cauchy asymptotic distribution.
{"title":"Gaussian Volterra processes: Asymptotic growth and statistical estimation","authors":"Y. Mishura, K. Ralchenko, S. Shklyar","doi":"10.1090/tpms/1190","DOIUrl":"https://doi.org/10.1090/tpms/1190","url":null,"abstract":"The paper is devoted to three-parametric self-similar Gaussian Volterra processes that generalize fractional Brownian motion. We study the asymptotic growth of such processes and the properties of long- and short-range dependence. Then we consider the problem of the drift parameter estimation for Ornstein–Uhlenbeck process driven by Gaussian Volterra process under consideration. We construct a strongly consistent estimator and investigate its asymptotic properties. Namely, we prove that it has the Cauchy asymptotic distribution.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44591218","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}
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.
{"title":"Revisiting recurrence criteria of birth and death processes. Short proofs","authors":"O. Zakusylo","doi":"10.1090/tpms/1182","DOIUrl":"https://doi.org/10.1090/tpms/1182","url":null,"abstract":"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 \u0000\u0000 \u0000 \u0000 \u0000 T\u0000 ∞\u0000 \u0000 \u0000 {T^infty }\u0000 \u0000\u0000 be the passage time from zero to \u0000\u0000 \u0000 ∞\u0000 infty\u0000 \u0000\u0000. The regularity criterion says that \u0000\u0000 \u0000 \u0000 \u0000 \u0000 T\u0000 ∞\u0000 \u0000 \u0000 >\u0000 ∞\u0000 \u0000 {T^infty } > infty\u0000 \u0000\u0000 if and only if \u0000\u0000 \u0000 \u0000 \u0000 E\u0000 \u0000 \u0000 \u0000 T\u0000 ∞\u0000 \u0000 \u0000 >\u0000 ∞\u0000 \u0000 mathbb {E}{T^infty } > infty\u0000 \u0000\u0000. 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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44596144","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}
A. Malyarenko, Y. Mishura, A. Olenko, M. Ostoja-Starzewski
{"title":"Editorial","authors":"A. Malyarenko, Y. Mishura, A. Olenko, M. Ostoja-Starzewski","doi":"10.1090/tpms/1183","DOIUrl":"https://doi.org/10.1090/tpms/1183","url":null,"abstract":"","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48138371","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}
<p>We show that an isotropic random field on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper U left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">SU(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not necessarily isotropic as a random field on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S cubed"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S cubed"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is necessarily a superposition of uncorrelated random harmonic homogeneous polynomials, such that the one of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is necessarily a superposition of uncorrelated random spin weighted functions of every possible spin weight in the range <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet minus StartFraction d Over 2 EndFraction comma ellipsis comma StartFraction d Over 2 EndFraction EndSet"> <mml:semantics> <mml:mrow> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-OPEN"> <mml:mo maxsize="1.2em" minsize="1.2em">{</mml:mo> </mml:mrow> </mml:mstyle> <mml:mo>−<!-- − --></mml:mo> <mml:mfrac> <mml:mi>d</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:mfrac> <mml:mi>d</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-CLOSE"> <mml:mo maxsize="1.2em" minsize="1.2em">}</mml:mo> </mml:mrow> </mml:mstyle> </mml:mrow> <mml:annotation encoding="application/x-tex">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
{"title":"Isotropic random spin weighted functions on 𝑆² vs isotropic random fields on 𝑆³","authors":"Michele Stecconi","doi":"10.1090/tpms/1177","DOIUrl":"https://doi.org/10.1090/tpms/1177","url":null,"abstract":"<p>We show that an isotropic random field on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S upper U left-parenthesis 2 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:mi>U</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">SU(2)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is not necessarily isotropic as a random field on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S cubed\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">S^3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, 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 <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S cubed\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">S^3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is necessarily a superposition of uncorrelated random harmonic homogeneous polynomials, such that the one of degree <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\">\u0000 <mml:semantics>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is necessarily a superposition of uncorrelated random spin weighted functions of every possible spin weight in the range <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartSet minus StartFraction d Over 2 EndFraction comma ellipsis comma StartFraction d Over 2 EndFraction EndSet\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mstyle scriptlevel=\"0\">\u0000 <mml:mrow class=\"MJX-TeXAtom-OPEN\">\u0000 <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">{</mml:mo>\u0000 </mml:mrow>\u0000 </mml:mstyle>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mfrac>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mfrac>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mo>…<!-- … --></mml:mo>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mfrac>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mfrac>\u0000 <mml:mstyle scriptlevel=\"0\">\u0000 <mml:mrow class=\"MJX-TeXAtom-CLOSE\">\u0000 <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">}</mml:mo>\u0000 </mml:mrow>\u0000 </mml:mstyle>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">bigl {","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46401291","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}
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.
{"title":"On spectral theory of random fields in the ball","authors":"N. Leonenko, A. Malyarenko, A. Olenko","doi":"10.1090/tpms/1175","DOIUrl":"https://doi.org/10.1090/tpms/1175","url":null,"abstract":"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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49597930","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}
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 .
{"title":"Spatiotemporal covariance functions for Laplacian ARMA fields in higher dimensions","authors":"G. Terdik","doi":"10.1090/tpms/1173","DOIUrl":"https://doi.org/10.1090/tpms/1173","url":null,"abstract":"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 \u0000\u0000 \u0000 \u0000 d\u0000 ≥\u0000 2\u0000 \u0000 dgeq 2\u0000 \u0000\u0000.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48298724","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}
<p>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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Subscript n"> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">X_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies the condition that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript x right-arrow normal infinity Endscripts StartFraction log upper H left-parenthesis x right-parenthesis Over left-parenthesis log x right-parenthesis Superscript delta Baseline EndFraction equals 0"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:munder> <mml:mfrac> <mml:mrow> <mml:mi>log</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>log</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>δ<!-- δ --></mml:mi> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">lim _{ xrightarrow infty } frac {log H(x)}{(log x)^delta } = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than delta greater-than 1 slash 2"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex
{"title":"On the other LIL for variables without finite variance","authors":"R. Pakshirajan, M. Sreehari","doi":"10.1090/tpms/1179","DOIUrl":"https://doi.org/10.1090/tpms/1179","url":null,"abstract":"<p>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 <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X Subscript n\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">X_n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> satisfies the condition that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"limit Underscript x right-arrow normal infinity Endscripts StartFraction log upper H left-parenthesis x right-parenthesis Over left-parenthesis log x right-parenthesis Superscript delta Baseline EndFraction equals 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:munder>\u0000 <mml:mo movablelimits=\"true\" form=\"prefix\">lim</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:munder>\u0000 <mml:mfrac>\u0000 <mml:mrow>\u0000 <mml:mi>log</mml:mi>\u0000 <mml:mo><!-- --></mml:mo>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>log</mml:mi>\u0000 <mml:mo><!-- --></mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mi>δ<!-- δ --></mml:mi>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 </mml:mfrac>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">lim _{ xrightarrow infty } frac {log H(x)}{(log x)^delta } = 0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for some <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0 greater-than delta greater-than 1 slash 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mi>δ<!-- δ --></mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46810295","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}
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.
{"title":"Stationary solutions of a second-order differential equation with operator coefficients","authors":"M. Horodnii","doi":"10.1090/tpms/1171","DOIUrl":"https://doi.org/10.1090/tpms/1171","url":null,"abstract":"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.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42362540","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}
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