An extension of Shannon’s entropy power inequality when one of the summands is Gaussian was provided by Costa in 1985, known as Costa’s concavity inequality. We consider the additive Gaussian noise channel with a more realistic assumption, i.e. the input and noise components are not independent and their dependence structure follows the well-known multivariate Gaussian copula. Two generalizations for the first- and second-order derivatives of the differential entropy of the output signal for dependent multivariate random variables are derived. It is shown that some previous results in the literature are particular versions of our results. Using these derivatives, concavity of the entropy power, under certain mild conditions, is proved. Finally, special one-dimensional versions of our general results are described which indeed reveal an extension of the one-dimensional case of Costa’s concavity inequality to the dependent case. An illustrative example is also presented.
{"title":"Costa’s concavity inequality for dependent variables based on the multivariate Gaussian copula","authors":"","doi":"10.1017/jpr.2022.128","DOIUrl":"https://doi.org/10.1017/jpr.2022.128","url":null,"abstract":"\u0000 An extension of Shannon’s entropy power inequality when one of the summands is Gaussian was provided by Costa in 1985, known as Costa’s concavity inequality. We consider the additive Gaussian noise channel with a more realistic assumption, i.e. the input and noise components are not independent and their dependence structure follows the well-known multivariate Gaussian copula. Two generalizations for the first- and second-order derivatives of the differential entropy of the output signal for dependent multivariate random variables are derived. It is shown that some previous results in the literature are particular versions of our results. Using these derivatives, concavity of the entropy power, under certain mild conditions, is proved. Finally, special one-dimensional versions of our general results are described which indeed reveal an extension of the one-dimensional case of Costa’s concavity inequality to the dependent case. An illustrative example is also presented.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44975318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-28eCollection Date: 2024-03-01DOI: 10.1055/s-0043-1767731
Katherine E Pendleton, Andres Hernandez-Garcia, Jennifer M Lyu, Ian M Campbell, Chad A Shaw, Julie Vogt, Frances A High, Patricia K Donahoe, Wendy K Chung, Daryl A Scott
FOXP1 encodes a transcription factor involved in tissue regulation and cell-type-specific functions. Haploinsufficiency of FOXP1 is associated with a neurodevelopmental disorder: autosomal dominant mental retardation with language impairment with or without autistic features. More recently, heterozygous FOXP1 variants have also been shown to cause a variety of structural birth defects including central nervous system (CNS) anomalies, congenital heart defects, congenital anomalies of the kidney and urinary tract, cryptorchidism, and hypospadias. In this report, we present a previously unpublished case of an individual with congenital diaphragmatic hernia (CDH) who carries an approximately 3.8 Mb deletion. Based on this deletion, and deletions previously reported in two other individuals with CDH, we define a CDH critical region on chromosome 3p13 that includes FOXP1 and four other protein-coding genes. We also provide detailed clinical descriptions of two previously reported individuals with CDH who carry de novo, pathogenic variants in FOXP1 that are predicted to trigger nonsense-mediated mRNA decay. A subset of individuals with putatively deleterious FOXP4 variants has also been shown to develop CDH. Since FOXP proteins function as homo- or heterodimers and the homologs of FOXP1 and FOXP4 are expressed at the same time points in the embryonic mouse diaphragm, they may function together as a dimer, or in parallel as homodimers, to regulate gene expression during diaphragm development. Not all individuals with heterozygous, loss-of-function changes in FOXP1 develop CDH. Hence, we conclude that FOXP1 acts as a susceptibility factor that contributes to the development of CDH in conjunction with other genetic, epigenetic, environmental, and/or stochastic factors.
{"title":"<i>FOXP1</i> Haploinsufficiency Contributes to the Development of Congenital Diaphragmatic Hernia.","authors":"Katherine E Pendleton, Andres Hernandez-Garcia, Jennifer M Lyu, Ian M Campbell, Chad A Shaw, Julie Vogt, Frances A High, Patricia K Donahoe, Wendy K Chung, Daryl A Scott","doi":"10.1055/s-0043-1767731","DOIUrl":"10.1055/s-0043-1767731","url":null,"abstract":"<p><p><i>FOXP1</i> encodes a transcription factor involved in tissue regulation and cell-type-specific functions. Haploinsufficiency of <i>FOXP1</i> is associated with a neurodevelopmental disorder: autosomal dominant mental retardation with language impairment with or without autistic features. More recently, heterozygous <i>FOXP1</i> variants have also been shown to cause a variety of structural birth defects including central nervous system (CNS) anomalies, congenital heart defects, congenital anomalies of the kidney and urinary tract, cryptorchidism, and hypospadias. In this report, we present a previously unpublished case of an individual with congenital diaphragmatic hernia (CDH) who carries an approximately 3.8 Mb deletion. Based on this deletion, and deletions previously reported in two other individuals with CDH, we define a CDH critical region on chromosome 3p13 that includes <i>FOXP1</i> and four other protein-coding genes. We also provide detailed clinical descriptions of two previously reported individuals with CDH who carry de novo, pathogenic variants in <i>FOXP1</i> that are predicted to trigger nonsense-mediated mRNA decay. A subset of individuals with putatively deleterious <i>FOXP4</i> variants has also been shown to develop CDH. Since FOXP proteins function as homo- or heterodimers and the homologs of <i>FOXP1</i> and <i>FOXP4</i> are expressed at the same time points in the embryonic mouse diaphragm, they may function together as a dimer, or in parallel as homodimers, to regulate gene expression during diaphragm development. Not all individuals with heterozygous, loss-of-function changes in <i>FOXP1</i> develop CDH. Hence, we conclude that <i>FOXP1</i> acts as a susceptibility factor that contributes to the development of CDH in conjunction with other genetic, epigenetic, environmental, and/or stochastic factors.</p>","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"41 1","pages":"29-34"},"PeriodicalIF":0.4,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10984716/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88458636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Guus Berkelmans, S. Bhulai, R. D. van der Mei, Joris Pries
Measuring and quantifying dependencies between random variables (RVs) can give critical insights into a dataset. Typical questions are: ‘Do underlying relationships exist?’, ‘Are some variables redundant?’, and ‘Is some target variable Y highly or weakly dependent on variable X?’ Interestingly, despite the evident need for a general-purpose measure of dependency between RVs, common practice is that most data analysts use the Pearson correlation coefficient to quantify dependence between RVs, while it is recognized that the correlation coefficient is essentially a measure for linear dependency only. Although many attempts have been made to define more generic dependency measures, there is no consensus yet on a standard, general-purpose dependency function. In fact, several ideal properties of a dependency function have been proposed, but without much argumentation. Motivated by this, we discuss and revise the list of desired properties and propose a new dependency function that meets all these requirements. This general-purpose dependency function provides data analysts with a powerful means to quantify the level of dependence between variables. To this end, we also provide Python code to determine the dependency function for use in practice.
{"title":"The Berkelmans–Pries dependency function: A generic measure of dependence between random variables","authors":"Guus Berkelmans, S. Bhulai, R. D. van der Mei, Joris Pries","doi":"10.1017/jpr.2022.118","DOIUrl":"https://doi.org/10.1017/jpr.2022.118","url":null,"abstract":"\u0000 Measuring and quantifying dependencies between random variables (RVs) can give critical insights into a dataset. Typical questions are: ‘Do underlying relationships exist?’, ‘Are some variables redundant?’, and ‘Is some target variable Y highly or weakly dependent on variable X?’ Interestingly, despite the evident need for a general-purpose measure of dependency between RVs, common practice is that most data analysts use the Pearson correlation coefficient to quantify dependence between RVs, while it is recognized that the correlation coefficient is essentially a measure for linear dependency only. Although many attempts have been made to define more generic dependency measures, there is no consensus yet on a standard, general-purpose dependency function. In fact, several ideal properties of a dependency function have been proposed, but without much argumentation. Motivated by this, we discuss and revise the list of desired properties and propose a new dependency function that meets all these requirements. This general-purpose dependency function provides data analysts with a powerful means to quantify the level of dependence between variables. To this end, we also provide Python code to determine the dependency function for use in practice.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48700970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider a class of processes describing a population consisting of k types of individuals. The process is almost surely absorbed at the origin within finite time, and we study the expected time taken for such extinction to occur. We derive simple and precise asymptotic estimates for this expected persistence time, starting either from a single individual or from a quasi-equilibrium state, in the limit as a system size parameter N tends to infinity. Our process need not be a Markov process on $ {mathbb Z}_+^k$ ; we allow the possibility that individuals’ lifetimes may follow more general distributions than the exponential distribution.
{"title":"Asymptotic persistence time formulae for multitype birth–death processes","authors":"F. Ball, D. Clancy","doi":"10.1017/jpr.2022.102","DOIUrl":"https://doi.org/10.1017/jpr.2022.102","url":null,"abstract":"Abstract We consider a class of processes describing a population consisting of k types of individuals. The process is almost surely absorbed at the origin within finite time, and we study the expected time taken for such extinction to occur. We derive simple and precise asymptotic estimates for this expected persistence time, starting either from a single individual or from a quasi-equilibrium state, in the limit as a system size parameter N tends to infinity. Our process need not be a Markov process on \u0000$ {mathbb Z}_+^k$\u0000 ; we allow the possibility that individuals’ lifetimes may follow more general distributions than the exponential distribution.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"60 1","pages":"895 - 920"},"PeriodicalIF":1.0,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44398001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We prove some estimates for the variations of transition probabilities of the (1+1)-affine process. From these estimates we deduce the strong Feller and the ergodic properties of the total variation distance of the process. The key strategy is the pathwise construction and analysis of several Markov couplings using strong solutions of stochastic equations.
{"title":"Strong feller and ergodic properties of the (1+1)-affine process","authors":"Shukai Chen, Zenghu Li","doi":"10.1017/jpr.2022.100","DOIUrl":"https://doi.org/10.1017/jpr.2022.100","url":null,"abstract":"Abstract We prove some estimates for the variations of transition probabilities of the (1+1)-affine process. From these estimates we deduce the strong Feller and the ergodic properties of the total variation distance of the process. The key strategy is the pathwise construction and analysis of several Markov couplings using strong solutions of stochastic equations.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"60 1","pages":"812 - 834"},"PeriodicalIF":1.0,"publicationDate":"2023-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48520342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The Bruss–Robertson–Steele (BRS) inequality bounds the expected number of items of random size which can be packed into a given suitcase. Remarkably, no independence assumptions are needed on the random sizes, which points to a simple explanation; the inequality is the integrated form of an $omega$ -by- $omega$ inequality, as this note proves.
{"title":"The Bruss–Robertson–Steele inequality","authors":"L. Rogers","doi":"10.1017/jpr.2022.122","DOIUrl":"https://doi.org/10.1017/jpr.2022.122","url":null,"abstract":"Abstract The Bruss–Robertson–Steele (BRS) inequality bounds the expected number of items of random size which can be packed into a given suitcase. Remarkably, no independence assumptions are needed on the random sizes, which points to a simple explanation; the inequality is the integrated form of an \u0000$omega$\u0000 -by- \u0000$omega$\u0000 inequality, as this note proves.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"60 1","pages":"1112 - 1114"},"PeriodicalIF":1.0,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46917918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We present an efficient algorithm to generate a discrete uniform distribution on a set of p elements using a biased random source for p prime. The algorithm generalizes Von Neumann’s method and improves the computational efficiency of Dijkstra’s method. In addition, the algorithm is extended to generate a discrete uniform distribution on any finite set based on the prime factorization of integers. The average running time of the proposed algorithm is overall sublinear: $operatorname{O}!(n/log n)$ .
{"title":"An efficient method for generating a discrete uniform distribution using a biased random source","authors":"Xiaoyu Lei","doi":"10.1017/jpr.2022.111","DOIUrl":"https://doi.org/10.1017/jpr.2022.111","url":null,"abstract":"Abstract We present an efficient algorithm to generate a discrete uniform distribution on a set of p elements using a biased random source for p prime. The algorithm generalizes Von Neumann’s method and improves the computational efficiency of Dijkstra’s method. In addition, the algorithm is extended to generate a discrete uniform distribution on any finite set based on the prime factorization of integers. The average running time of the proposed algorithm is overall sublinear: \u0000$operatorname{O}!(n/log n)$\u0000 .","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"60 1","pages":"1069 - 1078"},"PeriodicalIF":1.0,"publicationDate":"2023-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45110759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study a sceptical rumour model on the non-negative integer line. The model starts with two spreaders at sites 0, 1 and sceptical ignorants at all other natural numbers. Then each sceptic transmits the rumour, independently, to the individuals within a random distance on its right after s/he receives the rumour from at least two different sources. We say that the process survives if the size of the set of vertices which heard the rumour in this fashion is infinite. We calculate the probability of survival exactly, and obtain some bounds for the tail distribution of the final range of the rumour among sceptics. We also prove that the rumour dies out among non-sceptics and sceptics, under the same condition.
{"title":"On the probability of rumour survival among sceptics","authors":"N. Esmaeeli, F. Sajadi","doi":"10.1017/jpr.2022.113","DOIUrl":"https://doi.org/10.1017/jpr.2022.113","url":null,"abstract":"Abstract We study a sceptical rumour model on the non-negative integer line. The model starts with two spreaders at sites 0, 1 and sceptical ignorants at all other natural numbers. Then each sceptic transmits the rumour, independently, to the individuals within a random distance on its right after s/he receives the rumour from at least two different sources. We say that the process survives if the size of the set of vertices which heard the rumour in this fashion is infinite. We calculate the probability of survival exactly, and obtain some bounds for the tail distribution of the final range of the rumour among sceptics. We also prove that the rumour dies out among non-sceptics and sceptics, under the same condition.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"60 1","pages":"1096 - 1111"},"PeriodicalIF":1.0,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47865620","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let ${X_n}_{nin{mathbb{N}}}$ be an ${mathbb{X}}$ -valued iterated function system (IFS) of Lipschitz maps defined as $X_0 in {mathbb{X}}$ and for $ngeq 1$ , $X_n;:!=;F(X_{n-1},vartheta_n)$ , where ${vartheta_n}_{n ge 1}$ are independent and identically distributed random variables with common probability distribution $mathfrak{p}$ , $F(cdot,cdot)$ is Lipschitz continuous in the first variable, and $X_0$ is independent of ${vartheta_n}_{n ge 1}$ . Under parametric perturbation of both F and $mathfrak{p}$ , we are interested in the robustness of the V-geometrical ergodicity property of ${X_n}_{nin{mathbb{N}}}$ , of its invariant probability measure, and finally of the probability distribution of $X_n$ . Specifically, we propose a pattern of assumptions for studying such robustness properties for an IFS. This pattern is implemented for the autoregressive processes with autoregressive conditional heteroscedastic errors, and for IFS under roundoff error or under thresholding/truncation. Moreover, we provide a general set of assumptions covering the classical Feller-type hypotheses for an IFS to be a V-geometrical ergodic process. An accurate bound for the rate of convergence is also provided.
设${X_n}_{nin{mathbb{N}}}$为一个${mathbb{X}}$值的Lipschitz映射迭代函数系统(IFS)，定义为$X_0 in {mathbb{X}}$和$ngeq 1$, $X_n;:!=;F(X_{n-1},vartheta_n)$，其中${vartheta_n}_{n ge 1}$为具有共同概率分布的独立同分布随机变量，$mathfrak{p}$, $F(cdot,cdot)$在第一个变量上为Lipschitz连续，$X_0$独立于${vartheta_n}_{n ge 1}$。在F和$mathfrak{p}$的参数扰动下，我们感兴趣的是${X_n}_{nin{mathbb{N}}}$的v几何遍历性的鲁棒性，它的不变概率测度的鲁棒性，最后是$X_n$的概率分布的鲁棒性。具体来说，我们提出了一种假设模式来研究IFS的这种鲁棒性。该模式适用于具有自回归条件异方差误差的自回归过程，以及舍入误差或阈值/截断下的IFS。此外，我们还提供了一组一般假设，涵盖了经典的feller型假设，以证明IFS是一个v几何遍历过程。并给出了收敛速度的精确界。
{"title":"Robustness of iterated function systems of Lipschitz maps","authors":"L. Hervé, J. Ledoux","doi":"10.1017/jpr.2022.107","DOIUrl":"https://doi.org/10.1017/jpr.2022.107","url":null,"abstract":"Abstract Let \u0000${X_n}_{nin{mathbb{N}}}$\u0000 be an \u0000${mathbb{X}}$\u0000 -valued iterated function system (IFS) of Lipschitz maps defined as \u0000$X_0 in {mathbb{X}}$\u0000 and for \u0000$ngeq 1$\u0000 , \u0000$X_n;:!=;F(X_{n-1},vartheta_n)$\u0000 , where \u0000${vartheta_n}_{n ge 1}$\u0000 are independent and identically distributed random variables with common probability distribution \u0000$mathfrak{p}$\u0000 , \u0000$F(cdot,cdot)$\u0000 is Lipschitz continuous in the first variable, and \u0000$X_0$\u0000 is independent of \u0000${vartheta_n}_{n ge 1}$\u0000 . Under parametric perturbation of both F and \u0000$mathfrak{p}$\u0000 , we are interested in the robustness of the V-geometrical ergodicity property of \u0000${X_n}_{nin{mathbb{N}}}$\u0000 , of its invariant probability measure, and finally of the probability distribution of \u0000$X_n$\u0000 . Specifically, we propose a pattern of assumptions for studying such robustness properties for an IFS. This pattern is implemented for the autoregressive processes with autoregressive conditional heteroscedastic errors, and for IFS under roundoff error or under thresholding/truncation. Moreover, we provide a general set of assumptions covering the classical Feller-type hypotheses for an IFS to be a V-geometrical ergodic process. An accurate bound for the rate of convergence is also provided.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"60 1","pages":"921 - 944"},"PeriodicalIF":1.0,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46212452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The notion of ordered system signature, originally defined for independent and identical coherent systems, is first extended to the case of independent and non-identical coherent systems, and then some key properties that help simplify its computation are established. Through its use, a dynamic ordered system signature is defined next, which facilitates a systematic study of dynamic properties of several coherent systems under a life test. The theoretical results established here are then illustrated through some specific examples. Finally, the usefulness in the evaluation of aging used systems of the concepts introduced is demonstrated.
{"title":"On ordered system signature and its dynamic version for coherent systems with applications","authors":"He Yi, N. Balakrishnan, Xiang Li","doi":"10.1017/jpr.2022.110","DOIUrl":"https://doi.org/10.1017/jpr.2022.110","url":null,"abstract":"Abstract The notion of ordered system signature, originally defined for independent and identical coherent systems, is first extended to the case of independent and non-identical coherent systems, and then some key properties that help simplify its computation are established. Through its use, a dynamic ordered system signature is defined next, which facilitates a systematic study of dynamic properties of several coherent systems under a life test. The theoretical results established here are then illustrated through some specific examples. Finally, the usefulness in the evaluation of aging used systems of the concepts introduced is demonstrated.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"60 1","pages":"982 - 1002"},"PeriodicalIF":1.0,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45277978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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