Abstract Birth–death processes form a natural class where ideas and results on large deviations can be tested. We derive a large-deviation principle under an assumption that the rate of jump down (death) grows asymptotically linearly with the population size, while the rate of jump up (birth) grows sublinearly. We establish a large-deviation principle under various forms of scaling of the underlying process and the corresponding normalization of the logarithm of the large-deviation probabilities. The results show interesting features of dependence of the rate functional upon the parameters of the process and the forms of scaling and normalization.
{"title":"A large-deviation principle for birth–death processes with a linear rate of downward jumps","authors":"Artem Logachov, Yuri Suhov, Nikita Vvedenskaya, Anatoly Yambartsev","doi":"10.1017/jpr.2023.75","DOIUrl":"https://doi.org/10.1017/jpr.2023.75","url":null,"abstract":"Abstract Birth–death processes form a natural class where ideas and results on large deviations can be tested. We derive a large-deviation principle under an assumption that the rate of jump down (death) grows asymptotically linearly with the population size, while the rate of jump up (birth) grows sublinearly. We establish a large-deviation principle under various forms of scaling of the underlying process and the corresponding normalization of the logarithm of the large-deviation probabilities. The results show interesting features of dependence of the rate functional upon the parameters of the process and the forms of scaling and normalization.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135813748","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 In this paper, we time-change the generalized counting process (GCP) by an independent inverse mixed stable subordinator to obtain a fractional version of the GCP. We call it the mixed fractional counting process (MFCP). The system of fractional differential equations that governs its state probabilities is obtained using the Z transform method. Its one-dimensional distribution, mean, variance, covariance, probability generating function, and factorial moments are obtained. It is shown that the MFCP exhibits the long-range dependence property whereas its increment process has the short-range dependence property. As an application we consider a risk process in which the claims are modelled using the MFCP. For this risk process, we obtain an asymptotic behaviour of its finite-time ruin probability when the claim sizes are subexponentially distributed and the initial capital is arbitrarily large. Later, we discuss some distributional properties of a compound version of the GCP.
{"title":"On a time-changed variant of the generalized counting process","authors":"M. Khandakar, K. K. Kataria","doi":"10.1017/jpr.2023.70","DOIUrl":"https://doi.org/10.1017/jpr.2023.70","url":null,"abstract":"Abstract In this paper, we time-change the generalized counting process (GCP) by an independent inverse mixed stable subordinator to obtain a fractional version of the GCP. We call it the mixed fractional counting process (MFCP). The system of fractional differential equations that governs its state probabilities is obtained using the Z transform method. Its one-dimensional distribution, mean, variance, covariance, probability generating function, and factorial moments are obtained. It is shown that the MFCP exhibits the long-range dependence property whereas its increment process has the short-range dependence property. As an application we consider a risk process in which the claims are modelled using the MFCP. For this risk process, we obtain an asymptotic behaviour of its finite-time ruin probability when the claim sizes are subexponentially distributed and the initial capital is arbitrarily large. Later, we discuss some distributional properties of a compound version of the GCP.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136262230","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 in a general graph-theoretic formulation a long-range percolation model introduced by Lamperti [27]. For various underlying digraphs, we discuss connections between this model and random exchange processes. We clarify, for all $n in mathbb{N}$ , under which conditions the lattices $mathbb{N}_0^n$ and $mathbb{Z}^n$ are essentially covered in this model. Moreover, for all $n geq 2$ , we establish that it is impossible to cover the directed n -ary tree in our model.
摘要本文研究了Lamperti[27]引入的远程渗流模型的一般图论公式。对于各种底层有向图，我们讨论了该模型与随机交换过程之间的联系。我们澄清，对于所有$n in mathbb{N}$，在哪些条件下，网格$mathbb{N}_0^n$和$mathbb{Z}^n$基本上覆盖在这个模型中。此外，对于所有$n geq 2$，我们确定在我们的模型中不可能覆盖有向n元树。
{"title":"Boolean percolation on digraphs and random exchange processes","authors":"Georg Braun","doi":"10.1017/jpr.2023.76","DOIUrl":"https://doi.org/10.1017/jpr.2023.76","url":null,"abstract":"Abstract We study in a general graph-theoretic formulation a long-range percolation model introduced by Lamperti [27]. For various underlying digraphs, we discuss connections between this model and random exchange processes. We clarify, for all $n in mathbb{N}$ , under which conditions the lattices $mathbb{N}_0^n$ and $mathbb{Z}^n$ are essentially covered in this model. Moreover, for all $n geq 2$ , we establish that it is impossible to cover the directed n -ary tree in our model.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134972398","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 Measures of uncertainty are a topic of considerable and growing interest. Recently, the introduction of extropy as a measure of uncertainty, dual to Shannon entropy, has opened up interest in new aspects of the subject. Since there are many versions of entropy, a unified formulation has been introduced to work with all of them in an easy way. Here we consider the possibility of defining a unified formulation for extropy by introducing a measure depending on two parameters. For particular choices of parameters, this measure provides the well-known formulations of extropy. Moreover, the unified formulation of extropy is also analyzed in the context of the Dempster–Shafer theory of evidence, and an application to classification problems is given.
{"title":"The unified extropy and its versions in classical and Dempster–Shafer theories","authors":"Francesco Buono, Yong Deng, Maria Longobardi","doi":"10.1017/jpr.2023.68","DOIUrl":"https://doi.org/10.1017/jpr.2023.68","url":null,"abstract":"Abstract Measures of uncertainty are a topic of considerable and growing interest. Recently, the introduction of extropy as a measure of uncertainty, dual to Shannon entropy, has opened up interest in new aspects of the subject. Since there are many versions of entropy, a unified formulation has been introduced to work with all of them in an easy way. Here we consider the possibility of defining a unified formulation for extropy by introducing a measure depending on two parameters. For particular choices of parameters, this measure provides the well-known formulations of extropy. Moreover, the unified formulation of extropy is also analyzed in the context of the Dempster–Shafer theory of evidence, and an application to classification problems is given.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135368312","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 There are some connections between aging notions, stochastic orders, and expected utilities. It is known that the DRHR (decreasing reversed hazard rate) aging notion can be characterized via the comparative statics result of risk aversion, and that the location-independent riskier order preserves monotonicity between risk premium and the Arrow–Pratt measure of risk aversion, and that the dispersive order preserves this monotonicity for the larger class of increasing utilities. Here, the aging notions ILR (increasing likelihood ratio), IFR (increasing failure rate), IGLR (increasing generalized likelihood ratio), and IGFR (increasing generalized failure rate) are characterized in terms of expected utilities. Based on these observations, we recover the closure properties of ILR, IFR, and DRHR under convolution, and of IGLR and IGFR under product, and investigate the closure properties of the dispersive order, location-independent riskier order, excess wealth order, the total time on test transform order under convolution, and the star order under product. We have some new findings.
{"title":"Aging notions, stochastic orders, and expected utilities","authors":"Jianping Yang, Weiwei Zhuang, Taizhong Hu","doi":"10.1017/jpr.2023.71","DOIUrl":"https://doi.org/10.1017/jpr.2023.71","url":null,"abstract":"Abstract There are some connections between aging notions, stochastic orders, and expected utilities. It is known that the DRHR (decreasing reversed hazard rate) aging notion can be characterized via the comparative statics result of risk aversion, and that the location-independent riskier order preserves monotonicity between risk premium and the Arrow–Pratt measure of risk aversion, and that the dispersive order preserves this monotonicity for the larger class of increasing utilities. Here, the aging notions ILR (increasing likelihood ratio), IFR (increasing failure rate), IGLR (increasing generalized likelihood ratio), and IGFR (increasing generalized failure rate) are characterized in terms of expected utilities. Based on these observations, we recover the closure properties of ILR, IFR, and DRHR under convolution, and of IGLR and IGFR under product, and investigate the closure properties of the dispersive order, location-independent riskier order, excess wealth order, the total time on test transform order under convolution, and the star order under product. We have some new findings.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135884256","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 More than half a century ago, it was proved that the increasing failure rate (IFR) property is preserved under the formation of k -out-of- n systems (order statistics) when the lifetimes of the components are independent and have a common absolutely continuous distribution function. However, this property has not yet been proved in the discrete case. Here we give a proof based on the log-concavity property of the function $f({{mathrm{e}}}^x)$ . Furthermore, we extend this property to general distribution functions and general coherent systems under some conditions.
{"title":"Resolving an old problem on the preservation of the IFR property under the formation of -out-of- systems with discrete distributions","authors":"Mahdi Alimohammadi, Jorge Navarro","doi":"10.1017/jpr.2023.63","DOIUrl":"https://doi.org/10.1017/jpr.2023.63","url":null,"abstract":"Abstract More than half a century ago, it was proved that the increasing failure rate (IFR) property is preserved under the formation of k -out-of- n systems (order statistics) when the lifetimes of the components are independent and have a common absolutely continuous distribution function. However, this property has not yet been proved in the discrete case. Here we give a proof based on the log-concavity property of the function $f({{mathrm{e}}}^x)$ . Furthermore, we extend this property to general distribution functions and general coherent systems under some conditions.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136113869","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 Consider a well-shuffled deck of cards of n different types where each type occurs m times. In a complete feedback game, a player is asked to guess the top card from the deck. After each guess, the top card is revealed to the player and is removed from the deck. The total number of correct guesses in a complete feedback game has attracted significant interest in the past few decades. Under different regimes of m , n , the expected number of correct guesses, under the greedy (optimal) strategy, has been obtained by various authors, while there are not many results available about the fluctuations. In this paper we establish a central limit theorem with Berry–Esseen bounds when m is fixed and n is large. Our results extend to the case of decks where different types may have different multiplicity, under suitable assumptions.
{"title":"Central limit theorem in complete feedback games","authors":"Andrea Ottolini, Raghavendra Tripathi","doi":"10.1017/jpr.2023.64","DOIUrl":"https://doi.org/10.1017/jpr.2023.64","url":null,"abstract":"Abstract Consider a well-shuffled deck of cards of n different types where each type occurs m times. In a complete feedback game, a player is asked to guess the top card from the deck. After each guess, the top card is revealed to the player and is removed from the deck. The total number of correct guesses in a complete feedback game has attracted significant interest in the past few decades. Under different regimes of m , n , the expected number of correct guesses, under the greedy (optimal) strategy, has been obtained by various authors, while there are not many results available about the fluctuations. In this paper we establish a central limit theorem with Berry–Esseen bounds when m is fixed and n is large. Our results extend to the case of decks where different types may have different multiplicity, under suitable assumptions.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136113217","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 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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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