Abstract In the classical gambler’s ruin problem, the gambler plays an adversary with initial capitals z and $a-z$ , respectively, where $a>0$ and $0< z < a$ are integers. At each round, the gambler wins or loses a dollar with probabilities p and $1-p$ . The game continues until one of the two players is ruined. For even a and $0
在经典的赌徒破产问题中，赌徒的对手分别以大写字母z和$a-z$开头，其中$a>0$和$0< z < a$是整数。在每一轮中，赌徒以p和$1-p$的概率赢或输一美元。游戏继续进行，直到两个玩家中的一个被毁。对于偶数a和$0<zleq {a}/{2}$，以$p in [0,{frac{1}{2}}]$为索引的游戏持续时间(总回合数)的分布族显示为单调(增加)似然比，而对于${a}/{2} leq z<a$，以$p in [{frac{1}{2}}, 1]$为索引的持续时间的分布族具有单调(减少)似然比。特别是，对于$z={a}/{2}$，就似然比顺序而言，持续时间的分布通过$p={frac{1}{2}}$在$p in [0,1]$上最大化。奇数a的情况也考虑通常的随机顺序。此外，作为极限，简要讨论了布朗运动的第一退出时间。
{"title":"Stochastic ordering results on the duration of the gambler’s ruin game","authors":"Shoou-Ren Hsiau, Yi-Ching Yao","doi":"10.1017/jpr.2023.62","DOIUrl":"https://doi.org/10.1017/jpr.2023.62","url":null,"abstract":"Abstract In the classical gambler’s ruin problem, the gambler plays an adversary with initial capitals z and $a-z$ , respectively, where $a>0$ and $0< z < a$ are integers. At each round, the gambler wins or loses a dollar with probabilities p and $1-p$ . The game continues until one of the two players is ruined. For even a and $0<zleq {a}/{2}$ , the family of distributions of the duration (total number of rounds) of the game indexed by $p in [0,{frac{1}{2}}]$ is shown to have monotone (increasing) likelihood ratio, while for ${a}/{2} leq z<a$ , the family of distributions of the duration indexed by $p in [{frac{1}{2}}, 1]$ has monotone (decreasing) likelihood ratio. In particular, for $z={a}/{2}$ , in terms of the likelihood ratio order, the distribution of the duration is maximized over $p in [0,1]$ by $p={frac{1}{2}}$ . The case of odd a is also considered in terms of the usual stochastic order. Furthermore, as a limit, the first exit time of Brownian motion is briefly discussed.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135351786","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}
George V. Moustakides, Xujun Liu, Olgica Milenkovic
Abstract Candidates arrive sequentially for an interview process which results in them being ranked relative to their predecessors. Based on the ranks available at each time, a decision mechanism must be developed that selects or dismisses the current candidate in an effort to maximize the chance of selecting the best. This classical version of the ‘secretary problem’ has been studied in depth, mostly using combinatorial approaches, along with numerous other variants. We consider a particular new version where, during reviewing, it is possible to query an external expert to improve the probability of making the correct decision. Unlike existing formulations, we consider experts that are not necessarily infallible and may provide suggestions that can be faulty. For the solution of our problem we adopt a probabilistic methodology and view the querying times as consecutive stopping times which we optimize with the help of optimal stopping theory. For each querying time we must also design a mechanism to decide whether or not we should terminate the search at the querying time. This decision is straightforward under the usual assumption of infallible experts, but when experts are faulty it has a far more intricate structure.
{"title":"Optimal stopping methodology for the secretary problem with random queries","authors":"George V. Moustakides, Xujun Liu, Olgica Milenkovic","doi":"10.1017/jpr.2023.61","DOIUrl":"https://doi.org/10.1017/jpr.2023.61","url":null,"abstract":"Abstract Candidates arrive sequentially for an interview process which results in them being ranked relative to their predecessors. Based on the ranks available at each time, a decision mechanism must be developed that selects or dismisses the current candidate in an effort to maximize the chance of selecting the best. This classical version of the ‘secretary problem’ has been studied in depth, mostly using combinatorial approaches, along with numerous other variants. We consider a particular new version where, during reviewing, it is possible to query an external expert to improve the probability of making the correct decision. Unlike existing formulations, we consider experts that are not necessarily infallible and may provide suggestions that can be faulty. For the solution of our problem we adopt a probabilistic methodology and view the querying times as consecutive stopping times which we optimize with the help of optimal stopping theory. For each querying time we must also design a mechanism to decide whether or not we should terminate the search at the querying time. This decision is straightforward under the usual assumption of infallible experts, but when experts are faulty it has a far more intricate structure.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135790097","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 This paper analyzes the training process of generative adversarial networks (GANs) via stochastic differential equations (SDEs). It first establishes SDE approximations for the training of GANs under stochastic gradient algorithms, with precise error bound analysis. It then describes the long-run behavior of GAN training via the invariant measures of its SDE approximations under proper conditions. This work builds a theoretical foundation for GAN training and provides analytical tools to study its evolution and stability.
{"title":"Stochastic differential equation approximations of generative adversarial network training and its long-run behavior","authors":"Haoyang Cao, Xin Guo","doi":"10.1017/jpr.2023.57","DOIUrl":"https://doi.org/10.1017/jpr.2023.57","url":null,"abstract":"Abstract This paper analyzes the training process of generative adversarial networks (GANs) via stochastic differential equations (SDEs). It first establishes SDE approximations for the training of GANs under stochastic gradient algorithms, with precise error bound analysis. It then describes the long-run behavior of GAN training via the invariant measures of its SDE approximations under proper conditions. This work builds a theoretical foundation for GAN training and provides analytical tools to study its evolution and stability.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135831357","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 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":"25 1","pages":"0"},"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":"75 1","pages":"0"},"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":" ","pages":""},"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}
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":" ","pages":""},"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":" ","pages":""},"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}
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