This paper studies the performance of greedy matching algorithms on bipartite graphs [Formula: see text]. We focus primarily on three classical algorithms: [Formula: see text], which sequentially selects random edges from [Formula: see text]; [Formula: see text], which sequentially matches random vertices in [Formula: see text] to random neighbors; and [Formula: see text], which generates a random priority order over vertices in [Formula: see text] and then sequentially matches random vertices in [Formula: see text] to their highest-priority remaining neighbor. Prior work has focused on identifying the worst-case approximation ratio for each algorithm. This guarantee is highest for [Formula: see text] and lowest for [Formula: see text]. Our work instead studies the average performance of these algorithms when the edge set [Formula: see text] is random. Our first result compares [Formula: see text] and [Formula: see text] and shows that on average, [Formula: see text] produces more matches. This result holds for finite graphs (in contrast to previous asymptotic results) and also applies to “many to one” matching in which each vertex in [Formula: see text] can match with multiple vertices in [Formula: see text]. Our second result compares [Formula: see text] and [Formula: see text] and shows that the better worst-case guarantee of [Formula: see text] does not translate into better average performance. In “one to one” settings where each vertex in [Formula: see text] can match with only one vertex in [Formula: see text], the algorithms result in the same number of matches. When each vertex in [Formula: see text] can match with two vertices in [Formula: see text] produces more matches than [Formula: see text].
本文研究了贪心匹配算法在二部图上的性能[公式:见文]。我们主要关注三种经典算法:[公式:见文]，它依次从[公式:见文]中选择随机边缘;[公式:见文]，将[公式:见文]中的随机顶点顺序匹配到随机邻居;和[公式:见文]，它在[公式:见文]的顶点上生成一个随机的优先顺序，然后顺序地将[公式:见文]中的随机顶点与其剩余的最高优先级邻居匹配。先前的工作集中在确定每种算法的最坏情况近似比。这种保证对于[公式:见正文]是最高的，对于[公式:见正文]是最低的。我们的工作是研究当边缘集[公式:见文本]是随机时这些算法的平均性能。我们的第一个结果比较了[Formula: see text]和[Formula: see text]，结果显示平均而言，[Formula: see text]产生了更多的匹配。这个结果适用于有限图(与之前的渐近结果相反)，也适用于“多对一”匹配，其中[公式:见文本]中的每个顶点可以与[公式:见文本]中的多个顶点匹配。我们的第二个结果比较了[Formula: see text]和[Formula: see text]，结果表明[Formula: see text]更好的最坏情况保证并不能转化为更好的平均性能。在“一对一”设置中，[公式:见文本]中的每个顶点只能与[公式:见文本]中的一个顶点匹配，算法会产生相同数量的匹配。当[Formula: see text]中的每个顶点都能与[Formula: see text]中的两个顶点匹配时，产生的匹配数比[Formula: see text]多。
{"title":"Greedy Matching in Bipartite Random Graphs","authors":"N. Arnosti","doi":"10.1287/stsy.2021.0082","DOIUrl":"https://doi.org/10.1287/stsy.2021.0082","url":null,"abstract":"This paper studies the performance of greedy matching algorithms on bipartite graphs [Formula: see text]. We focus primarily on three classical algorithms: [Formula: see text], which sequentially selects random edges from [Formula: see text]; [Formula: see text], which sequentially matches random vertices in [Formula: see text] to random neighbors; and [Formula: see text], which generates a random priority order over vertices in [Formula: see text] and then sequentially matches random vertices in [Formula: see text] to their highest-priority remaining neighbor. Prior work has focused on identifying the worst-case approximation ratio for each algorithm. This guarantee is highest for [Formula: see text] and lowest for [Formula: see text]. Our work instead studies the average performance of these algorithms when the edge set [Formula: see text] is random. Our first result compares [Formula: see text] and [Formula: see text] and shows that on average, [Formula: see text] produces more matches. This result holds for finite graphs (in contrast to previous asymptotic results) and also applies to “many to one” matching in which each vertex in [Formula: see text] can match with multiple vertices in [Formula: see text]. Our second result compares [Formula: see text] and [Formula: see text] and shows that the better worst-case guarantee of [Formula: see text] does not translate into better average performance. In “one to one” settings where each vertex in [Formula: see text] can match with only one vertex in [Formula: see text], the algorithms result in the same number of matches. When each vertex in [Formula: see text] can match with two vertices in [Formula: see text] produces more matches than [Formula: see text].","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66531612","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}
Momentum stochastic gradient descent (MSGD) algorithm has been widely applied to many nonconvex optimization problems in machine learning (e.g., training deep neural networks, variational Bayesian inference, etc.). Despite its empirical success, there is still a lack of theoretical understanding of convergence properties of MSGD. To fill this gap, we propose to analyze the algorithmic behavior of MSGD by diffusion approximations for nonconvex optimization problems with strict saddle points and isolated local optima. Our study shows that the momentum helps escape from saddle points but hurts the convergence within the neighborhood of optima (if without the step size annealing or momentum annealing). Our theoretical discovery partially corroborates the empirical success of MSGD in training deep neural networks.
{"title":"A Diffusion Approximation Theory of Momentum Stochastic Gradient Descent in Nonconvex Optimization","authors":"Tianyi Liu, Zhehui Chen, Enlu Zhou, T. Zhao","doi":"10.1287/stsy.2021.0083","DOIUrl":"https://doi.org/10.1287/stsy.2021.0083","url":null,"abstract":"Momentum stochastic gradient descent (MSGD) algorithm has been widely applied to many nonconvex optimization problems in machine learning (e.g., training deep neural networks, variational Bayesian inference, etc.). Despite its empirical success, there is still a lack of theoretical understanding of convergence properties of MSGD. To fill this gap, we propose to analyze the algorithmic behavior of MSGD by diffusion approximations for nonconvex optimization problems with strict saddle points and isolated local optima. Our study shows that the momentum helps escape from saddle points but hurts the convergence within the neighborhood of optima (if without the step size annealing or momentum annealing). Our theoretical discovery partially corroborates the empirical success of MSGD in training deep neural networks.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42964439","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 considers the problem of optimally controlling the drift of a Brownian motion with a finite set of possible drift rates so as to minimize the long-run average cost, consisting of fixed costs for changing the drift rate, processing costs for maintaining the drift rate, holding costs on the state of the process, and costs for instantaneous controls to keep the process within a prescribed range. We show that, under mild assumptions on the processing costs and the fixed costs for changing the drift rate, there is a strongly ordered optimal policy, that is, an optimal policy that limits the use of each drift rate to a single interval; when the process reaches the upper limit of that interval, the policy either changes to the next lower drift rate deterministically or resorts to instantaneous controls to keep the process within the prescribed range, and when the process reaches the lower limit of the interval, the policy either changes to the next higher drift rate deterministically or again resorts to instantaneous controls to keep the process within the prescribed range. We prove the optimality of such a policy by constructing smooth relative value functions satisfying the associated simplified optimality criteria. This paper shows that, under the proportional changeover cost assumption, each drift rate is active in at most one contiguous range and that the transitions between drift rates are strongly ordered. The results reduce the complexity of proving the optimality of such a policy by proving the existence of optimal relative value functions that constitute a nondecreasing sequence of functions. As a consequence, the constructive arguments lead to a practical procedure for solving the problem that is tens of thousands of times faster than previously reported methods.
{"title":"Average Cost Brownian Drift Control with Proportional Changeover Costs","authors":"John H. Vande Vate","doi":"10.1287/stsy.2021.0071","DOIUrl":"https://doi.org/10.1287/stsy.2021.0071","url":null,"abstract":"This paper considers the problem of optimally controlling the drift of a Brownian motion with a finite set of possible drift rates so as to minimize the long-run average cost, consisting of fixed costs for changing the drift rate, processing costs for maintaining the drift rate, holding costs on the state of the process, and costs for instantaneous controls to keep the process within a prescribed range. We show that, under mild assumptions on the processing costs and the fixed costs for changing the drift rate, there is a strongly ordered optimal policy, that is, an optimal policy that limits the use of each drift rate to a single interval; when the process reaches the upper limit of that interval, the policy either changes to the next lower drift rate deterministically or resorts to instantaneous controls to keep the process within the prescribed range, and when the process reaches the lower limit of the interval, the policy either changes to the next higher drift rate deterministically or again resorts to instantaneous controls to keep the process within the prescribed range. We prove the optimality of such a policy by constructing smooth relative value functions satisfying the associated simplified optimality criteria. This paper shows that, under the proportional changeover cost assumption, each drift rate is active in at most one contiguous range and that the transitions between drift rates are strongly ordered. The results reduce the complexity of proving the optimality of such a policy by proving the existence of optimal relative value functions that constitute a nondecreasing sequence of functions. As a consequence, the constructive arguments lead to a practical procedure for solving the problem that is tens of thousands of times faster than previously reported methods.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48242353","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 journal is pleased to publish the abstracts of the winner and finalists of the 2020 Applied Probability Society’s student paper competition. The 2020 student paper prize committee was chaired by Amy Ward. The 2020 committee members are (in alphabetical order by last name): Alessandro Arlotto, Sayan Banerjee, Junfei Huang, Jefferson Huang, Rouba Ibrahim, Peter Jacko, Henry Lam, Nan Liu, Yunan Liu, Siva Theja Maguluri, Giang Nguyen, Mariana Olvera-Cravioto, Lerzan Örmeci, Erhun Özkan, Jamol Pender, Weina Wang, Amy Ward (chair), Linwei Xin, Kuang Xu, Galit Yom-Tov, Assaf Zeevi, Jiheng Zhang, Zeyu Zheng, Yuan Zhong, Enlu Zhou, and Serhan Ziya.
{"title":"Applied Probability Society Student Paper Competition: Abstracts of 2020 Finalists","authors":"","doi":"10.1287/stsy.2021.0069","DOIUrl":"https://doi.org/10.1287/stsy.2021.0069","url":null,"abstract":"The journal is pleased to publish the abstracts of the winner and finalists of the 2020 Applied Probability Society’s student paper competition. The 2020 student paper prize committee was chaired by Amy Ward. The 2020 committee members are (in alphabetical order by last name): Alessandro Arlotto, Sayan Banerjee, Junfei Huang, Jefferson Huang, Rouba Ibrahim, Peter Jacko, Henry Lam, Nan Liu, Yunan Liu, Siva Theja Maguluri, Giang Nguyen, Mariana Olvera-Cravioto, Lerzan Örmeci, Erhun Özkan, Jamol Pender, Weina Wang, Amy Ward (chair), Linwei Xin, Kuang Xu, Galit Yom-Tov, Assaf Zeevi, Jiheng Zhang, Zeyu Zheng, Yuan Zhong, Enlu Zhou, and Serhan Ziya.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43001238","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}
We study in this paper large-time asymptotics of the empirical vector associated with a family of finite-state mean-field systems with multiclasses. The empirical vector is composed of local empirical measures characterizing the different classes within the system. As the number of particles in the system goes to infinity, the empirical vector process converges toward the solution to a McKean-Vlasov system. First, we investigate the large deviations principles of the invariant distribution from the limiting McKean-Vlasov system. Then, we examine the metastable phenomena arising at a large scale and large time. Finally, we estimate the rate of convergence of the empirical vector process to its invariant measure. Given the local homogeneity in the system, our results are established in a product space.
{"title":"Large-Time Behavior of Finite-State Mean-Field Systems With Multiclasses","authors":"D. Dawson, Ahmed Sid-Ali, Yiqiang Q. Zhao","doi":"10.1287/stsy.2022.0100","DOIUrl":"https://doi.org/10.1287/stsy.2022.0100","url":null,"abstract":"We study in this paper large-time asymptotics of the empirical vector associated with a family of finite-state mean-field systems with multiclasses. The empirical vector is composed of local empirical measures characterizing the different classes within the system. As the number of particles in the system goes to infinity, the empirical vector process converges toward the solution to a McKean-Vlasov system. First, we investigate the large deviations principles of the invariant distribution from the limiting McKean-Vlasov system. Then, we examine the metastable phenomena arising at a large scale and large time. Finally, we estimate the rate of convergence of the empirical vector process to its invariant measure. Given the local homogeneity in the system, our results are established in a product space.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49084936","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 uses the generator comparison approach of Stein’s method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. The “standard” generator comparison approach starts with the Poisson equation for the diffusion, and the main technical difficulty is to obtain bounds on the derivatives of the solution to the Poisson equation, also known as Stein factor bounds. In this paper we propose starting with the Poisson equation of the Markov chain; we term this the prelimit approach. Although one still needs Stein factor bounds, they now correspond to finite differences of the Markov chain Poisson equation solution rather than the derivatives of the solution to the diffusion Poisson equation. In certain cases, the former are easier to obtain. We use the [Formula: see text] model as a simple working example to illustrate our approach.
{"title":"The Prelimit Generator Comparison Approach of Stein’s Method","authors":"Anton Braverman","doi":"10.1287/stsy.2021.0085","DOIUrl":"https://doi.org/10.1287/stsy.2021.0085","url":null,"abstract":"This paper uses the generator comparison approach of Stein’s method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. The “standard” generator comparison approach starts with the Poisson equation for the diffusion, and the main technical difficulty is to obtain bounds on the derivatives of the solution to the Poisson equation, also known as Stein factor bounds. In this paper we propose starting with the Poisson equation of the Markov chain; we term this the prelimit approach. Although one still needs Stein factor bounds, they now correspond to finite differences of the Markov chain Poisson equation solution rather than the derivatives of the solution to the diffusion Poisson equation. In certain cases, the former are easier to obtain. We use the [Formula: see text] model as a simple working example to illustrate our approach.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49591485","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}
M. Jonckheere, P. Moyal, Claudia Ram'irez, N. Soprano-Loto
We consider a matching system where items arrive one by one at each node of a compatibility network according to Poisson processes and depart from it as soon as they are matched to a compatible item. The matching policy considered is a generalized max-weight policy where decisions can be noisy. Additionally, some of the nodes may have impatience, that is, leave the system before being matched. Using specific properties of the max-weight policy, we construct a simple quadratic Lyapunov function. This allows us to establish stability results, to prove exponential convergence speed toward the stationary measure, and to give explicit bounds for the stationary mean of the largest queue size in the system. We finally illustrate some of these results using simulations on toy examples.
{"title":"Generalized Max-Weight Policies in Stochastic Matching","authors":"M. Jonckheere, P. Moyal, Claudia Ram'irez, N. Soprano-Loto","doi":"10.1287/stsy.2022.0098","DOIUrl":"https://doi.org/10.1287/stsy.2022.0098","url":null,"abstract":"We consider a matching system where items arrive one by one at each node of a compatibility network according to Poisson processes and depart from it as soon as they are matched to a compatible item. The matching policy considered is a generalized max-weight policy where decisions can be noisy. Additionally, some of the nodes may have impatience, that is, leave the system before being matched. Using specific properties of the max-weight policy, we construct a simple quadratic Lyapunov function. This allows us to establish stability results, to prove exponential convergence speed toward the stationary measure, and to give explicit bounds for the stationary mean of the largest queue size in the system. We finally illustrate some of these results using simulations on toy examples.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46818283","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}
Sonja Otten, Ruslan K. Krenzler, H. Daduna, K. Kruse
We consider exponential single server queues with state-dependent arrival and service rates that evolve under influences of external environments. The transitions of the queues are influenced by the environment’s state and the movements of the environment depend on the status of the queues (bidirectional interaction). The environment is constructed in a way to encompass various models from the recent Operations Research literature, where a queue is coupled with an inventory or with reliability issues. With a Markovian joint queueing-environment process, we prove separability for a large class of such interactive systems; that is, the steady state distribution is of product form and explicitly given. The queue and the environment processes decouple asymptotically and in steady state. For nonseparable systems, we develop ergodicity and exponential ergodicity criteria via Lyapunov functions. By examples we explain principles for bounding departure rates of served customers (throughputs) of nonseparable systems by throughputs of related separable systems as upper and lower bound.
{"title":"Exponential Single Server Queues in an Interactive Random Environment","authors":"Sonja Otten, Ruslan K. Krenzler, H. Daduna, K. Kruse","doi":"10.1287/stsy.2023.0106","DOIUrl":"https://doi.org/10.1287/stsy.2023.0106","url":null,"abstract":"We consider exponential single server queues with state-dependent arrival and service rates that evolve under influences of external environments. The transitions of the queues are influenced by the environment’s state and the movements of the environment depend on the status of the queues (bidirectional interaction). The environment is constructed in a way to encompass various models from the recent Operations Research literature, where a queue is coupled with an inventory or with reliability issues. With a Markovian joint queueing-environment process, we prove separability for a large class of such interactive systems; that is, the steady state distribution is of product form and explicitly given. The queue and the environment processes decouple asymptotically and in steady state. For nonseparable systems, we develop ergodicity and exponential ergodicity criteria via Lyapunov functions. By examples we explain principles for bounding departure rates of served customers (throughputs) of nonseparable systems by throughputs of related separable systems as upper and lower bound.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48599022","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}
In this paper, we propose and analyze models of self policing in online communities, in which assessment activities, typically handled by firm employees, are shifted to the “crowd.” Our underlying objective is to maximize firm value by maintaining the quality of the online community to prevent attrition, which, given a parsimonious model of voter participation, we show can be achieved by efficiently utilizing the crowd of volunteer voters. To do so, we focus on minimizing the number of voters needed for each assessment, subject to service-level constraints, which depends on a voting aggregation rule. We focus our attention on classes of voting aggregators that are simple, interpretable, and implementable, which increases the chance of adoption in practice. We consider static and dynamic variants of simple majority-rule voting, with which each vote is treated equally. We also study static and dynamic variants of a more sophisticated voting rule that allows more accurate voters to have a larger influence in determining the aggregate decision. We consider both independent and correlated voters and show that correlation is detrimental to performance. Finally, we take a system view and characterize the limit of a costless crowdvoting system that relies solely on volunteer voters. If this limit does not satisfy target service levels, then costly firm employees are needed to supplement the crowd.
{"title":"Crowdvoting Judgment: An Analysis of Modern Peer Review","authors":"Michael R. Wagner","doi":"10.1287/stsy.2019.0053","DOIUrl":"https://doi.org/10.1287/stsy.2019.0053","url":null,"abstract":"In this paper, we propose and analyze models of self policing in online communities, in which assessment activities, typically handled by firm employees, are shifted to the “crowd.” Our underlying objective is to maximize firm value by maintaining the quality of the online community to prevent attrition, which, given a parsimonious model of voter participation, we show can be achieved by efficiently utilizing the crowd of volunteer voters. To do so, we focus on minimizing the number of voters needed for each assessment, subject to service-level constraints, which depends on a voting aggregation rule. We focus our attention on classes of voting aggregators that are simple, interpretable, and implementable, which increases the chance of adoption in practice. We consider static and dynamic variants of simple majority-rule voting, with which each vote is treated equally. We also study static and dynamic variants of a more sophisticated voting rule that allows more accurate voters to have a larger influence in determining the aggregate decision. We consider both independent and correlated voters and show that correlation is detrimental to performance. Finally, we take a system view and characterize the limit of a costless crowdvoting system that relies solely on volunteer voters. If this limit does not satisfy target service levels, then costly firm employees are needed to supplement the crowd.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/stsy.2019.0053","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47714906","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}
We study shot noise processes when the shot noises are weakly dependent, satisfying the ρ-mixing condition. We prove a functional weak law of large numbers and a functional central limit theorem for this shot noise process in an asymptotic regime with a high intensity of shots. The deterministic fluid limit is unaffected by the presence of weak dependence. The limit in the diffusion scale is a continuous Gaussian process whose covariance function explicitly captures the dependence among the noises. The model and results can be applied in financial and insurance risks with dependent claims as well as queueing systems with dependent service times. To prove the existence of the limit process, we employ the existence criterion that uses a maximal inequality requiring a set function with a superadditivity property. We identify such a set function for the limit process by exploiting the ρ-mixing condition. To prove the weak convergence, we establish the tightness property and the convergence of finite dimensional distributions. To prove tightness, we construct two auxiliary processes and apply an Ottaviani-type inequality for weakly dependent sequences.
{"title":"Functional Limit Theorems for Shot Noise Processes with Weakly Dependent Noises","authors":"G. Pang, Yuhang Zhou","doi":"10.1287/stsy.2019.0051","DOIUrl":"https://doi.org/10.1287/stsy.2019.0051","url":null,"abstract":"We study shot noise processes when the shot noises are weakly dependent, satisfying the ρ-mixing condition. We prove a functional weak law of large numbers and a functional central limit theorem for this shot noise process in an asymptotic regime with a high intensity of shots. The deterministic fluid limit is unaffected by the presence of weak dependence. The limit in the diffusion scale is a continuous Gaussian process whose covariance function explicitly captures the dependence among the noises. The model and results can be applied in financial and insurance risks with dependent claims as well as queueing systems with dependent service times. To prove the existence of the limit process, we employ the existence criterion that uses a maximal inequality requiring a set function with a superadditivity property. We identify such a set function for the limit process by exploiting the ρ-mixing condition. To prove the weak convergence, we establish the tightness property and the convergence of finite dimensional distributions. To prove tightness, we construct two auxiliary processes and apply an Ottaviani-type inequality for weakly dependent sequences.","PeriodicalId":36337,"journal":{"name":"Stochastic Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/stsy.2019.0051","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42747297","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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