Abstract We study the $R_beta$ -positivity and the existence of zero-temperature limits for a sequence of infinite-volume Gibbs measures $(mu_{beta}(!cdot!))_{beta geq 0}$ at inverse temperature $beta$ associated to a family of nearest-neighbor matrices $(Q_{beta})_{beta geq 0}$ reflected at the origin. We use a probabilistic approach based on the continued fraction theory previously introduced in Ferrari and Martínez (1993) and sharpened in Littin and Martínez (2010). Some necessary and sufficient conditions are provided to ensure (i) the existence of a unique infinite-volume Gibbs measure for large but finite values of $beta$ , and (ii) the existence of weak limits as $beta to infty$ . Some application examples are revised to put in context the main results of this work.
摘要研究了一组在原点反射的最近邻矩阵$(Q_{beta})_{beta geq 0}$的无穷体积Gibbs测量序列$(mu_{beta}(!cdot!))_{beta geq 0}$在逆温度$beta$的$R_beta$ -正性和零温度极限的存在性。我们使用了一种基于连续分数理论的概率方法，该理论先前在Ferrari和Martínez(1993)中引入，并在Littin和Martínez(2010)中得到了加强。给出了保证(1)对于大而有限值的$beta$存在唯一的无限体积Gibbs测度和(2)弱极限$beta to infty$存在的充分必要条件。修改了一些应用实例，以便将本工作的主要结果放在上下文中。
{"title":"<i>R</i>-positivity and the existence of zero-temperature limits of Gibbs measures on nearest-neighbor matrices","authors":"Jorge Littin Curinao, Gerardo Corredor Rincón","doi":"10.1017/jpr.2023.59","DOIUrl":"https://doi.org/10.1017/jpr.2023.59","url":null,"abstract":"Abstract We study the $R_beta$ -positivity and the existence of zero-temperature limits for a sequence of infinite-volume Gibbs measures $(mu_{beta}(!cdot!))_{beta geq 0}$ at inverse temperature $beta$ associated to a family of nearest-neighbor matrices $(Q_{beta})_{beta geq 0}$ reflected at the origin. We use a probabilistic approach based on the continued fraction theory previously introduced in Ferrari and Martínez (1993) and sharpened in Littin and Martínez (2010). Some necessary and sufficient conditions are provided to ensure (i) the existence of a unique infinite-volume Gibbs measure for large but finite values of $beta$ , and (ii) the existence of weak limits as $beta to infty$ . Some application examples are revised to put in context the main results of this work.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135816835","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 A comparison theorem for state-dependent regime-switching diffusion processes is established, which enables us to pathwise-control the evolution of the state-dependent switching component simply by Markov chains. Moreover, a sharp estimate on the stability of Markovian regime-switching processes under the perturbation of transition rate matrices is provided. Our approach is based on elaborate constructions of switching processes in the spirit of Skorokhod’s representation theorem varying according to the problem being dealt with. In particular, this method can cope with switching processes in an infinite state space and not necessarily of birth–death type. As an application, some known results on the ergodicity and stability of state-dependent regime-switching processes can be improved.
{"title":"Comparison theorem and stability under perturbation of transition rate matrices for regime-switching processes","authors":"Jinghai Shao","doi":"10.1017/jpr.2023.54","DOIUrl":"https://doi.org/10.1017/jpr.2023.54","url":null,"abstract":"Abstract A comparison theorem for state-dependent regime-switching diffusion processes is established, which enables us to pathwise-control the evolution of the state-dependent switching component simply by Markov chains. Moreover, a sharp estimate on the stability of Markovian regime-switching processes under the perturbation of transition rate matrices is provided. Our approach is based on elaborate constructions of switching processes in the spirit of Skorokhod’s representation theorem varying according to the problem being dealt with. In particular, this method can cope with switching processes in an infinite state space and not necessarily of birth–death type. As an application, some known results on the ergodicity and stability of state-dependent regime-switching processes can be improved.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134911852","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}
In this paper, several linear two-dimensional consecutive k-type systems are studied, which include the linear connected-(k, r)-out-of- $(m,n)colon! F$ system and the linear l-connected-(k, r)-out-of- $(m,n)colon! F$ system without/with overlapping. Reliabilities of these systems are studied via the finite Markov chain imbedding approach (FMCIA) in a novel way. Some numerical examples are provided to illustrate the theoretical results established here and also to demonstrate the efficiency of the developed method. Finally, some possible applications and generalizations of the developed results are pointed out.
{"title":"Reliability analyses of linear two-dimensional consecutive k-type systems","authors":"He Yi, N. Balakrishnan, Xiang Li","doi":"10.1017/jpr.2023.51","DOIUrl":"https://doi.org/10.1017/jpr.2023.51","url":null,"abstract":"\u0000 In this paper, several linear two-dimensional consecutive k-type systems are studied, which include the linear connected-(k, r)-out-of-\u0000 \u0000 \u0000 \u0000$(m,n)colon! F$\u0000\u0000 \u0000 system and the linear l-connected-(k, r)-out-of-\u0000 \u0000 \u0000 \u0000$(m,n)colon! F$\u0000\u0000 \u0000 system without/with overlapping. Reliabilities of these systems are studied via the finite Markov chain imbedding approach (FMCIA) in a novel way. Some numerical examples are provided to illustrate the theoretical results established here and also to demonstrate the efficiency of the developed method. Finally, some possible applications and generalizations of the developed results are pointed out.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43227810","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}
{"title":"JPR volume 60 issue 3 Cover and Back matter","authors":"","doi":"10.1017/jpr.2023.42","DOIUrl":"https://doi.org/10.1017/jpr.2023.42","url":null,"abstract":"","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43006388","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}
{"title":"JPR volume 60 issue 3 Cover and Front matter","authors":"","doi":"10.1017/jpr.2023.41","DOIUrl":"https://doi.org/10.1017/jpr.2023.41","url":null,"abstract":"","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47078496","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}
We consider parallel single-server queues in heavy traffic with randomly split Hawkes arrival processes. The service times are assumed to be independent and identically distributed (i.i.d.) in each queue and are independent in different queues. In the critically loaded regime at each queue, it is shown that the diffusion-scaled queueing and workload processes converge to a multidimensional reflected Brownian motion in the non-negative orthant with orthonormal reflections. For the model with abandonment, we also show that the corresponding limit is a multidimensional reflected Ornstein–Uhlenbeck diffusion in the non-negative orthant.
{"title":"Heavy-traffic limits for parallel single-server queues with randomly split Hawkes arrival processes","authors":"Bo Li, G. Pang","doi":"10.1017/jpr.2023.50","DOIUrl":"https://doi.org/10.1017/jpr.2023.50","url":null,"abstract":"\u0000 We consider parallel single-server queues in heavy traffic with randomly split Hawkes arrival processes. The service times are assumed to be independent and identically distributed (i.i.d.) in each queue and are independent in different queues. In the critically loaded regime at each queue, it is shown that the diffusion-scaled queueing and workload processes converge to a multidimensional reflected Brownian motion in the non-negative orthant with orthonormal reflections. For the model with abandonment, we also show that the corresponding limit is a multidimensional reflected Ornstein–Uhlenbeck diffusion in the non-negative orthant.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45371790","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}
We consider the problem of controlling the drift and diffusion rate of the endowment processes of two firms such that the joint survival probability is maximized. We assume that the endowment processes are continuous diffusions, driven by independent Brownian motions, and that the aggregate endowment is a Brownian motion with constant drift and diffusion rate. Our results reveal that the maximal joint survival probability depends only on the aggregate risk-adjusted return and on the maximal risk-adjusted return that can be implemented in each firm. Here the risk-adjusted return is understood as the drift rate divided by the squared diffusion rate.
{"title":"On the joint survival probability of two collaborating firms","authors":"S. Ankirchner, R. Hesse, Maike Klein","doi":"10.1017/jpr.2023.46","DOIUrl":"https://doi.org/10.1017/jpr.2023.46","url":null,"abstract":"\u0000 We consider the problem of controlling the drift and diffusion rate of the endowment processes of two firms such that the joint survival probability is maximized. We assume that the endowment processes are continuous diffusions, driven by independent Brownian motions, and that the aggregate endowment is a Brownian motion with constant drift and diffusion rate. Our results reveal that the maximal joint survival probability depends only on the aggregate risk-adjusted return and on the maximal risk-adjusted return that can be implemented in each firm. Here the risk-adjusted return is understood as the drift rate divided by the squared diffusion rate.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42352603","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}
We study the problem of detecting the community structure from the generalized stochastic block model with two communities (G2-SBM). Based on analysis of the Stieljtes transform of the empirical spectral distribution, we prove a Baik–Ben Arous–Péché (BBP)-type transition for the largest eigenvalue of the G2-SBM. For specific models, such as a hidden community model and an unbalanced stochastic block model, we provide precise formulas for the two largest eigenvalues, establishing the gap in the BBP-type transition.
{"title":"Phase transition for the generalized two-community stochastic block model","authors":"Sunmin Lee, J. Lee","doi":"10.1017/jpr.2023.44","DOIUrl":"https://doi.org/10.1017/jpr.2023.44","url":null,"abstract":"\u0000 We study the problem of detecting the community structure from the generalized stochastic block model with two communities (G2-SBM). Based on analysis of the Stieljtes transform of the empirical spectral distribution, we prove a Baik–Ben Arous–Péché (BBP)-type transition for the largest eigenvalue of the G2-SBM. For specific models, such as a hidden community model and an unbalanced stochastic block model, we provide precise formulas for the two largest eigenvalues, establishing the gap in the BBP-type transition.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42081494","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}
R. Cavazos-Cadena, H. Cruz-Suárez, Raúl Montes-de-Oca
This work concerns Markov decision chains on a denumerable state space endowed with a bounded cost function. The performance of a control policy is assessed by a long-run average criterion as measured by a risk-seeking decision maker with constant risk-sensitivity. Besides standard continuity–compactness conditions, the framework of the paper is determined by the following conditions: (i) the state process is communicating under each stationary policy, and (ii) the simultaneous Doeblin condition holds. Within this framework it is shown that (i) the optimal superior and inferior limit average value functions coincide and are constant, and (ii) the optimal average cost is characterized via an extended version of the Collatz–Wielandt formula in the theory of positive matrices.
{"title":"Characterization of the optimal average cost in Markov decision chains driven by a risk-seeking controller","authors":"R. Cavazos-Cadena, H. Cruz-Suárez, Raúl Montes-de-Oca","doi":"10.1017/jpr.2023.40","DOIUrl":"https://doi.org/10.1017/jpr.2023.40","url":null,"abstract":"\u0000 This work concerns Markov decision chains on a denumerable state space endowed with a bounded cost function. The performance of a control policy is assessed by a long-run average criterion as measured by a risk-seeking decision maker with constant risk-sensitivity. Besides standard continuity–compactness conditions, the framework of the paper is determined by the following conditions: (i) the state process is communicating under each stationary policy, and (ii) the simultaneous Doeblin condition holds. Within this framework it is shown that (i) the optimal superior and inferior limit average value functions coincide and are constant, and (ii) the optimal average cost is characterized via an extended version of the Collatz–Wielandt formula in the theory of positive matrices.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49013363","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}
We prove existence and uniqueness for the inverse-first-passage time problem for soft-killed Brownian motion using rather elementary methods relying on basic results from probability theory only. We completely avoid the relation to a suitable partial differential equation via a suitable Feynman–Kac representation, which was previously one of the main tools.
{"title":"An elementary approach to the inverse first-passage-time problem for soft-killed Brownian motion","authors":"Alexander Klump, Martin Kolb","doi":"10.1017/jpr.2023.39","DOIUrl":"https://doi.org/10.1017/jpr.2023.39","url":null,"abstract":"\u0000 We prove existence and uniqueness for the inverse-first-passage time problem for soft-killed Brownian motion using rather elementary methods relying on basic results from probability theory only. We completely avoid the relation to a suitable partial differential equation via a suitable Feynman–Kac representation, which was previously one of the main tools.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43098605","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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