This article is a survey of results concerning an inequality, which may be seen as a versatile tool to solve problems in the domain of Applied Probability. The inequality, which we call BRS-inequality, gives a convenient upper bound for the expected maximum number of non-negative random variables one can sum up without exceeding a given upper bound s > 0. One valuable property of the BRS-inequality is that it is valid without any hypothesis about independence of the random variables. Another welcome feature is that, once one sees that one can use it in a given problem, its application is often straightforward or not very involved. This survey is focussed, and we hope that it is pleasant and inspiring to read. Focus is easy to achieve, given that the BRS-inequality and its most useful versions can be displayed in five Theorems and their proofs. We try to present these in an appealing way. The objective to be inspiring is harder, and the best we can think of is offering a variety of applications. Our examples include comparisons between sums of i.i.d. versus non-identically distributed and/or dependent random variables, problems of condensing point processes, awkward processes, monotone subsequence problems, knapsack problems, online algorithms, tiling policies, Borel-Cantelli type problems, up to applications in the theory of resource dependent branching processes. Apart from our wish to present the inequality in an organised way, the motivation for this survey is the hope that interested readers may see potential of the inequality for their own problems. MSC2020 subject classifications: Primary 60-01; secondary 60-02.
{"title":"The BRS-inequality and its applications","authors":"F. Bruss","doi":"10.1214/20-PS351","DOIUrl":"https://doi.org/10.1214/20-PS351","url":null,"abstract":"This article is a survey of results concerning an inequality, which may be seen as a versatile tool to solve problems in the domain of Applied Probability. The inequality, which we call BRS-inequality, gives a convenient upper bound for the expected maximum number of non-negative random variables one can sum up without exceeding a given upper bound s > 0. One valuable property of the BRS-inequality is that it is valid without any hypothesis about independence of the random variables. Another welcome feature is that, once one sees that one can use it in a given problem, its application is often straightforward or not very involved. This survey is focussed, and we hope that it is pleasant and inspiring to read. Focus is easy to achieve, given that the BRS-inequality and its most useful versions can be displayed in five Theorems and their proofs. We try to present these in an appealing way. The objective to be inspiring is harder, and the best we can think of is offering a variety of applications. Our examples include comparisons between sums of i.i.d. versus non-identically distributed and/or dependent random variables, problems of condensing point processes, awkward processes, monotone subsequence problems, knapsack problems, online algorithms, tiling policies, Borel-Cantelli type problems, up to applications in the theory of resource dependent branching processes. Apart from our wish to present the inequality in an organised way, the motivation for this survey is the hope that interested readers may see potential of the inequality for their own problems. MSC2020 subject classifications: Primary 60-01; secondary 60-02.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"18 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66085439","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 survey aspects of prediction theory in infinitely many dimensions, with a view to the theory and applications of functional time series.
从函数时间序列的理论和应用的角度，研究了无限多维预测理论的各个方面。
{"title":"Prediction theory for stationary functional time series","authors":"N. Bingham","doi":"10.1214/20-ps360","DOIUrl":"https://doi.org/10.1214/20-ps360","url":null,"abstract":"We survey aspects of prediction theory in infinitely many dimensions, with a view to the theory and applications of functional time series.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44183465","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 article, we propose a modified technique for finding Stein operator for the class of infinitely divisible distributions using its characteristic function that relaxes the assumption of the first finite moment. Using this technique, we reproduce the Stein operators for stable distributions with $alphain(0,2)$ with less efforts. We have shown that a single approach with minor modifications is enough to solve the Stein equations for the stable distributions with $alphain(0,1)$ and $alphain(1,2)$. Finally, we give applications of our results for stable approximations.
{"title":"A unified approach to Stein’s method for stable distributions","authors":"N. S. Upadhye, K. Barman","doi":"10.1214/20-ps354","DOIUrl":"https://doi.org/10.1214/20-ps354","url":null,"abstract":"In this article, we propose a modified technique for finding Stein operator for the class of infinitely divisible distributions using its characteristic function that relaxes the assumption of the first finite moment. Using this technique, we reproduce the Stein operators for stable distributions with $alphain(0,2)$ with less efforts. We have shown that a single approach with minor modifications is enough to solve the Stein equations for the stable distributions with $alphain(0,1)$ and $alphain(1,2)$. Finally, we give applications of our results for stable approximations.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48660020","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 is devoted to tangent martingales in Banach spaces. We provide the definition of tangency through local characteristics, basic $L^p$- and $phi$-estimates, a precise construction of a decoupled tangent martingale, new estimates for vector-valued stochastic integrals, and several other claims concerning tangent martingales and local characteristics in infinite dimensions. This work extends various real-valued and vector-valued results in this direction e.g. due to Grigelionis, Hitczenko, Jacod, Kallenberg, Kwapien, McConnell, and Woyczynski. The vast majority of the assertions presented in the paper is done under the sufficient and necessary UMD assumption on the corresponding Banach space.
{"title":"Local characteristics and tangency of vector-valued martingales","authors":"I. Yaroslavtsev","doi":"10.1214/19-ps337","DOIUrl":"https://doi.org/10.1214/19-ps337","url":null,"abstract":"This paper is devoted to tangent martingales in Banach spaces. We provide the definition of tangency through local characteristics, basic $L^p$- and $phi$-estimates, a precise construction of a decoupled tangent martingale, new estimates for vector-valued stochastic integrals, and several other claims concerning tangent martingales and local characteristics in infinite dimensions. This work extends various real-valued and vector-valued results in this direction e.g. due to Grigelionis, Hitczenko, Jacod, Kallenberg, Kwapien, McConnell, and Woyczynski. The vast majority of the assertions presented in the paper is done under the sufficient and necessary UMD assumption on the corresponding Banach space.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45257304","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}
G. Bouzianis, L. Hughston, S. Jaimungal, Leandro S'anchez-Betancourt
We propose a class of financial models in which the prices of assets are Levy-Ito processes driven by Brownian motion and a dynamic Poisson random measure. Each such model consists of a pricing kernel, a money market account, and one or more risky assets. The Poisson random measure is associated with an $n$-dimensional Levy process. We show that the excess rate of return of a risky asset in a pure-jump model is given by an integral of the product of a term representing the riskiness of the asset and a term representing the level of market risk aversion. The integral is over the state space of the Poisson random measure and is taken with respect to the Levy measure associated with the $n$-dimensional Levy process. The resulting framework is applied to the theory of interest rates and foreign exchange, allowing one to construct new models as well as various generalizations of familiar models.
{"title":"Lévy-Ito models in finance","authors":"G. Bouzianis, L. Hughston, S. Jaimungal, Leandro S'anchez-Betancourt","doi":"10.1214/21-PS1","DOIUrl":"https://doi.org/10.1214/21-PS1","url":null,"abstract":"We propose a class of financial models in which the prices of assets are Levy-Ito processes driven by Brownian motion and a dynamic Poisson random measure. Each such model consists of a pricing kernel, a money market account, and one or more risky assets. The Poisson random measure is associated with an $n$-dimensional Levy process. We show that the excess rate of return of a risky asset in a pure-jump model is given by an integral of the product of a term representing the riskiness of the asset and a term representing the level of market risk aversion. The integral is over the state space of the Poisson random measure and is taken with respect to the Levy measure associated with the $n$-dimensional Levy process. The resulting framework is applied to the theory of interest rates and foreign exchange, allowing one to construct new models as well as various generalizations of familiar models.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44436326","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 develop a unified approach for solving a wide class of sequential selection problems. This class includes, but is not limited to, selection problems with no-information, rank-dependent rewards, and considers both fixed as well as random problem horizons. The proposed framework is based on a reduction of the original selection problem to one of optimal stopping for a sequence of judiciously constructed independent random variables. We demonstrate that our approach allows exact and efficient computation of optimal policies and various performance metrics thereof for a variety of sequential selection problems, several of which have not been solved to date.
{"title":"A unified approach for solving sequential selection problems","authors":"A. Goldenshluger, Y. Malinovsky, A. Zeevi","doi":"10.1214/19-ps333","DOIUrl":"https://doi.org/10.1214/19-ps333","url":null,"abstract":"In this paper we develop a unified approach for solving a wide class of sequential selection problems. This class includes, but is not limited to, selection problems with no-information, rank-dependent rewards, and considers both fixed as well as random problem horizons. The proposed framework is based on a reduction of the original selection problem to one of optimal stopping for a sequence of judiciously constructed independent random variables. We demonstrate that our approach allows exact and efficient computation of optimal policies and various performance metrics thereof for a variety of sequential selection problems, several of which have not been solved to date.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47211747","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 analyze the equilibrium properties of a large class of stochastic processes describing the fundamental biological process within bacterial cells, {em the production process of proteins}. Stochastic models classically used in this context to describe the time evolution of the numbers of mRNAs and proteins are presented and discussed. An extension of these models, which includes elongation phases of mRNAs and proteins, is introduced. A convergence result to equilibrium for the process associated to the number of proteins and mRNAs is proved and a representation of this equilibrium as a functional of a Poisson process in an extended state space is obtained. Explicit expressions for the first two moments of the number of mRNAs and proteins at equilibrium are derived, generalizing some classical formulas. Approximations used in the biological literature for the equilibrium distribution of the number of proteins are discussed and investigated in the light of these results. Several convergence results for the distribution of the number of proteins at equilibrium are in particular obtained under different scaling assumptions.
{"title":"Mathematical models of gene expression","authors":"Philippe Robert","doi":"10.1214/19-ps332","DOIUrl":"https://doi.org/10.1214/19-ps332","url":null,"abstract":"In this paper we analyze the equilibrium properties of a large class of stochastic processes describing the fundamental biological process within bacterial cells, {em the production process of proteins}. Stochastic models classically used in this context to describe the time evolution of the numbers of mRNAs and proteins are presented and discussed. An extension of these models, which includes elongation phases of mRNAs and proteins, is introduced. A convergence result to equilibrium for the process associated to the number of proteins and mRNAs is proved and a representation of this equilibrium as a functional of a Poisson process in an extended state space is obtained. Explicit expressions for the first two moments of the number of mRNAs and proteins at equilibrium are derived, generalizing some classical formulas. Approximations used in the biological literature for the equilibrium distribution of the number of proteins are discussed and investigated in the light of these results. Several convergence results for the distribution of the number of proteins at equilibrium are in particular obtained under different scaling assumptions.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49494633","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 quadratic variation of Gaussian processes plays an important rolein both stochastic analysis and in applications such as estimation ofmodel parameters, and for this reason the topic has been extensivelystudied in the literature. In this article we study the convergence ofquadratic sums of general Gaussian sequences. We provide necessary andsufficient conditions for different types of convergence includingconvergence in probability, almost sure convergence, $L^{p}$-convergenceas well as weak convergence. We use a practical and simple approachwhich simplifies the existing methodology considerably. As anapplication, we show how convergence of the quadratic variation of agiven process can be obtained by an appropriate choice of the underlyingsequence.
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