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Best Strategy for Each Team in The Regular Season to Win Champion in The Knockout Tournament 每支球队在常规赛中赢得淘汰赛冠军的最佳策略
arXiv: Probability Pub Date : 2020-06-02 DOI: 10.1137/20s1340460
Zijie Zhou
{"title":"Best Strategy for Each Team in The Regular Season to Win Champion in The Knockout Tournament","authors":"Zijie Zhou","doi":"10.1137/20s1340460","DOIUrl":"https://doi.org/10.1137/20s1340460","url":null,"abstract":"In J. Schwenk.(2018) ['What is the Correct Way to Seed a Knockout Tournament?' Retrieved from The American Mathematical Monthly], Schwenk identified a surprising weakness in the standard method of seeding a single elimination (or knockout) tournament. In particular, he showed that for a certain probability model for the outcomes of games it can be the case that the top seeded team would be less likely to win the tournament than the second seeded team. This raises the possibility that in certain situations it might be advantageous for a team to intentionally lose a game in an attempt to get a more optimal (though possibly lower) seed in the tournament. We examine this question in the context of a four team league which consists of a round robin \"regular season\" followed by a single elimination tournament with seedings determined by the results from the regular season [4]. Using the same probability model as Schwenk we show that there are situations where it is indeed optimal for a team to intentionally lose. Moreover, we show how a team can make the decision as to whether or not it should intentionally lose. We did two detailed analysis. One is for the situation where other teams always try to win every game. The other is for the situation where other teams are smart enough, namely they can also lose some games intentionally if necessary. The analysis involves computations in both probability and (multi-player) game theory.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80771446","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}
引用次数: 0
Single jump filtrations and local martingales 单跳滤波和局部鞅
arXiv: Probability Pub Date : 2020-06-01 DOI: 10.15559/20-VMSTA153
A. Gushchin
{"title":"Single jump filtrations and local martingales","authors":"A. Gushchin","doi":"10.15559/20-VMSTA153","DOIUrl":"https://doi.org/10.15559/20-VMSTA153","url":null,"abstract":"A single jump filtration $({mathscr{F}}_t)_{tin mathbb{R}_+}$ generated by a random variable $gamma$ with values in $overline{mathbb{R}}_+$ on a probability space $(Omega ,{mathscr{F}},mathsf{P})$ is defined as follows: a set $Ain {mathscr{F}}$ belongs to ${mathscr{F}}_t$ if $Acap {gamma >t}$ is either $varnothing$ or ${gamma >t}$. A process $M$ is proved to be a local martingale with respect to this filtration if and only if it has a representation $M_t=F(t){mathbb{1}}_{{t 0}$. This result seems to be new even in a special case that has been studied in the literature, namely, where ${mathscr{F}}$ is the smallest $sigma$-field with respect to which $gamma$ is measurable (and then the filtration is the smallest one with respect to which $gamma$ is a stopping time). As a consequence, a full description of all local martingales is given and they are classified according to their global behaviour.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78095595","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}
引用次数: 2
Study on A Direct Construction of the Standard Brownian Motion 标准布朗运动的一种直接构造研究
arXiv: Probability Pub Date : 2020-06-01 DOI: 10.9734/BPI/TPMCS/V1/5056D
Lo Gane Samb, N. Babacar, S. Harouna
{"title":"Study on A Direct Construction of the Standard Brownian Motion","authors":"Lo Gane Samb, N. Babacar, S. Harouna","doi":"10.9734/BPI/TPMCS/V1/5056D","DOIUrl":"https://doi.org/10.9734/BPI/TPMCS/V1/5056D","url":null,"abstract":"In this note, we combine the two approaches of Billingsley (1998) and Csőrgő and Revesz (1980), to provide a detailed sequential and descriptive for creating s standard Brownian motion, from a Brownian motion whose time space is the class of non-negative dyadic numbers. By adding the proof of Etemadi's inequality to text, it becomes self-readable and serves as an independent source for researches and professors.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73343467","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}
引用次数: 0
Irregular barrier reflected BDSDEs with general jumps under stochastic Lipschitz and linear growth conditions 在随机Lipschitz和线性生长条件下，不规则势垒反映了具有普遍跳跃的BDSDEs
arXiv: Probability Pub Date : 2020-06-01 DOI: 10.15559/20-VMSTA155
M. Marzougue, Yaya Sagna
{"title":"Irregular barrier reflected BDSDEs with general jumps under stochastic Lipschitz and linear growth conditions","authors":"M. Marzougue, Yaya Sagna","doi":"10.15559/20-VMSTA155","DOIUrl":"https://doi.org/10.15559/20-VMSTA155","url":null,"abstract":"In this paper, a solution is given to reflected backward doubly stochastic differential equations when the barrier is not necessarily right-continuous, and the noise is driven by two independent Brownian motions and an independent Poisson random measure. The existence and uniqueness of the solution is shown, firstly when the coefficients are stochastic Lipschitz, and secondly by weakening the conditions on the stochastic growth coefficient.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84734560","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}
引用次数: 2
Homological Percolation: The Formation of Giant k-Cycles 同调渗流:巨k环的形成
arXiv: Probability Pub Date : 2020-05-28 DOI: 10.1093/imrn/rnaa305
O. Bobrowski, P. Skraba
{"title":"Homological Percolation: The Formation of Giant k-Cycles","authors":"O. Bobrowski, P. Skraba","doi":"10.1093/imrn/rnaa305","DOIUrl":"https://doi.org/10.1093/imrn/rnaa305","url":null,"abstract":"In this paper we introduce and study a higher-dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant k-dimensional cycles (with 0-cycles being connected components). Considering a continuum percolation model in the flat d-dimensional torus, we show that all the giant k-cycles (k=1,...,d-1) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant k-cycles are increasing in k and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91042134","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}
引用次数: 14
Backward semi-martingales into Burgers turbulence 向后半鞅进入汉堡湍流
arXiv: Probability Pub Date : 2020-05-28 DOI: 10.1063/5.0036721
Florent Nzissila, O. Moutsinga, Fulgence Eyi Obiang
{"title":"Backward semi-martingales into Burgers turbulence","authors":"Florent Nzissila, O. Moutsinga, Fulgence Eyi Obiang","doi":"10.1063/5.0036721","DOIUrl":"https://doi.org/10.1063/5.0036721","url":null,"abstract":"In fluid dynamics governed by the one dimensional inviscid Burgers equation $partial_t u+upartial_x(u)=0$, the stirring is explained by the sticky particles model. A Markov process $([Z^1_t,Z^2_t],,tgeq0)$ describes the motion of random turbulent intervals which evolve inside an other Markov process $([Z^3_t,Z^4_t],,tgeq0)$, describing the motion of random clusters concerned with the turbulence. Then, the four velocity processes $(u(Z^i_t,t),,tgeq0)$ are backward semi-martingales.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73517639","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}
引用次数: 0
Almost sure behavior of linearly edge-reinforced random walks on the half-line 半线上线性边增强随机漫步的几乎确定行为
arXiv: Probability Pub Date : 2020-05-22 DOI: 10.1214/21-ejp674
Masato Takei
{"title":"Almost sure behavior of linearly edge-reinforced random walks on the half-line","authors":"Masato Takei","doi":"10.1214/21-ejp674","DOIUrl":"https://doi.org/10.1214/21-ejp674","url":null,"abstract":"We study linearly edge-reinforced random walks on $mathbb{Z}_+$, where each edge ${x,x+1}$ has the initial weight $x^{alpha} vee 1$, and each time an edge is traversed, its weight is increased by $Delta$. It is known that the walk is recurrent if and only if $alpha leq 1$. The aim of this paper is to study the almost sure behavior of the walk in the recurrent regime. For $alpha 0$, we obtain a limit theorem which is a counterpart of the law of the iterated logarithm for simple random walks. This reveals that the speed of the walk with $Delta>0$ is much slower than $Delta=0$. In the critical case $alpha=1$, our (almost sure) bounds for the trajectory of the walk shows that there is a phase transition of the speed at $Delta=2$.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82986976","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}
引用次数: 1
Convoluted Fractional Poisson Process 卷积分数泊松过程
arXiv: Probability Pub Date : 2020-05-21 DOI: 10.30757/ALEA.v18-46
K. K. Kataria, M. Khandakar
{"title":"Convoluted Fractional Poisson Process","authors":"K. K. Kataria, M. Khandakar","doi":"10.30757/ALEA.v18-46","DOIUrl":"https://doi.org/10.30757/ALEA.v18-46","url":null,"abstract":"In this paper, we introduce and study a convoluted version of the time fractional Poisson process by taking the discrete convolution with respect to space variable in the system of fractional differential equations that governs its state probabilities. We call the introduced process as the convoluted fractional Poisson process (CFPP). The explicit expression for the Laplace transform of its state probabilities are obtained whose inversion yields its one-dimensional distribution. Some of its statistical properties such as probability generating function, moment generating function, moments etc. are obtained. A special case of CFPP, namely, the convoluted Poisson process (CPP) is studied and its time-changed subordination relationships with CFPP are discussed. It is shown that the CPP is a Levy process using which the long-range dependence property of CFPP is established. Moreover, we show that the increments of CFPP exhibits short-range dependence property.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80027484","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}
引用次数: 7
Regularity properties of jump diffusions with irregular coefficients. 不规则系数跳跃扩散的正则性。
arXiv: Probability Pub Date : 2020-05-21 DOI: 10.1016/J.JMAA.2021.125220
Guohuan Zhao
{"title":"Regularity properties of jump diffusions with irregular coefficients.","authors":"Guohuan Zhao","doi":"10.1016/J.JMAA.2021.125220","DOIUrl":"https://doi.org/10.1016/J.JMAA.2021.125220","url":null,"abstract":"","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89529092","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}
引用次数: 3
Distribution-Dependent Stochastic Differential Delay Equations in finite and infinite dimensions 有限维和无限维的分布相关随机微分延迟方程
arXiv: Probability Pub Date : 2020-05-15 DOI: 10.1142/s0219025720500241
Rico Heinemann
{"title":"Distribution-Dependent Stochastic Differential Delay Equations in finite and infinite dimensions","authors":"Rico Heinemann","doi":"10.1142/s0219025720500241","DOIUrl":"https://doi.org/10.1142/s0219025720500241","url":null,"abstract":"We prove that distribution dependent (also called McKean--Vlasov) stochastic delay equations of the form begin{equation*} mathrm{d}X(t)= b(t,X_t,mathcal{L}_{X_t})mathrm{d}t+ sigma(t,X_t,mathcal{L}_{X_t})mathrm{d}W(t) end{equation*} have unique (strong) solutions in finite as well as infinite dimensional state spaces if the coefficients fulfill certain monotonicity assumptions.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84702942","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}
引用次数: 3
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